Mr Cook, who joined the technology giant in 1998 and has been its CEO for nearly 15 years after succeeding the late Steve Jobs, will take up the role of executive chairman.
He will remain in his current role until 1 September, when Mr Ternus, the current head of the hardware engineering department, will officially take up the position.
The move comes following months of speculation that Apple was searching for a successor to Mr Cook, with Mr Ternus seen as a likely candidate to take over.
In his new role, Mr Cook will "assist with certain aspects of the company, including engaging with policymakers around the world", Apple said.
In a statement, Mr Cook said: "It has been the greatest privilege of my life to be the CEO of Apple and to have been trusted to lead such an extraordinary company.
"I love Apple with all of my being, and I am so grateful to have had the opportunity to work with a team of such ingenious, innovative, creative, and deeply caring people who have been unwavering in their dedication to enriching the lives of our customers and creating the best products and services in the world."
Mr Cook described his successor as a "visionary" with "the soul of an innovator".
He added: "He [Ternus] is without question the right person to lead Apple into the future. I could not be more confident in his abilities and his character."
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Mr Ternus joined Apple in 2001, and during his 25 years at the company has been involved in a number of its major product releases, including the iPad and AirPods, as well as several generations of iPhone, Mac, and Apple Watch.
He referred to Mr Cook as his mentor and said he was "profoundly grateful" for the opportunity to "carry Apple's mission forward".
"I am filled with optimism about what we can achieve in the years to come," he added.
It happened at the Teotihuacan pyramids outside Mexico City, with footage showing the man firing from the top of one of the pyramids as people sought cover.
Authorities said two more Canadians were wounded, alongside a Russian and two Colombians, and that the attacker killed himself after Monday's shooting.
Two further people were injured in falls amid the panic and taken to hospital.
Officials also said they found a gun, a knife and ammunition after the shooting.
"What happened today in Teotihuacan pains us deeply," said Mexican President Claudia Sheinbaum on social media.
"I express my deepest sympathy to those affected and their families."
The attacker's identity has not yet been released.
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Teotihuacan, built between the first and seventh centuries, is one of Mexico's most popular tourist attractions and had 1.8 million visitors last year.
A tour guide, who spoke on condition of anonymity so as not to put his job at risk, told the Associated Press he was leading a group down one of the pyramids at around 11.30am local time when the gunman began to shoot.
A number of people stumbled in fear and fell down the stairs, he added.
He said he heard between 20 and 30 gunshots ring out before security officials arrived.
Canada's foreign affairs minister, Anita Anand, also expressed her sympathies to those affected by the shooting and said her thoughts were with the family of the woman who was killed.
The Cathy Newman Show, launching on 27 April at 7pm, combines agenda-setting interviews, exclusive investigations and spirited debate from Sky News' Westminster studio.
The show, airing Monday to Thursday on Sky News and YouTube, is designed to cut through the noise and make sense of the stories that matter.
'The perfect time to get started'
Newman, who joins Sky News after two decades at Channel 4, will lead a show featuring uninterrupted conversations that challenge perspectives – with a warm but rigorous approach.
The show will include interviews with leading political, public and cultural figures, original reporting and investigations, and analysis of the biggest UK, US and European stories.
"With the local elections coming up, it feels like the perfect time to get started," said Cathy Newman.
"I'm looking forward to welcoming guests on to the orange velvet sofa for big interviews, exclusives and real conversations that help make sense of it all. Join me from next week for a friendly but forensic chat."
There will also be longer-form investigations and documentaries that sit alongside the show, with a new podcast launching this autumn.
The programme is part of Sky News' 2030 strategy for premium, video-first journalism, designed to bring personality-led reporting to audiences wherever they are.
"Cathy's journalism is forensic, fearless and engaging, and this new format brings that to life in a way that truly connects with audiences," said Jonathan Levy, executive editor and managing director.
Join us from 27 April at 7pm for interviews that probe, investigations that reveal, and conversations that matter.
Valentina Gomez was due to speak at the Unite the Kingdom rally, which is being organised by far-right activist Tommy Robinson, on 16 May, and posted on X that her UK electronic travel authorisation (ETA) had been approved.
But after backlash from MPs and campaigners, the government has excluded the US-based Christian influencer because her presence "would not be conducive to the public good", Sky News understands.
This is the same reason given for Kanye West's permission to travel to the UK to headline the Wireless festival in London being blocked by the government. West had been heavily criticised in the past for antisemitic remarks.
After Ms Gomez was banned from entering the UK, she wrote on X: "I'm coming to England on a boat. They can try to ban me, but they cannot ban the TRUTH. See you May 16th."
In an accompanying video she racially abuses Home Secretary Shabana Mahmood and claims her ban is "because I'm not coming to rape or kill little girls".
The influencer uses social media to share her anti-Islam views and has previously posted a video of herself burning a Koran with a flamethrower.
The government has discretion to ban foreign nationals from the UK if their presence is not considered "conducive to the public good".
A ban can be ordered by the home secretary personally, and government guidance states it will normally involve serious issues such as national security, war crimes, corruption or extremism.
Sky News understands that in Ms Gomez's case, the government blocked her entry on the grounds that the democratic right to expression does not extend to promoting hatred or extremism.
The 26-year-old influencer attended and spoke at a previous Unite the Kingdom rally, to which between 110,000 and 150,000 people turned up.
She told the told the crowd at the event in September: "If these rapist Muslims take over they will not only rape your women, they will behead your sons."
The Muslim Council of Britain wrote to Ms Mahmood expressing concern about allowing Ms Gomez to enter the UK, saying to do so would "lead to less safety and security on the streets of Britain".
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The council has welcomed the decision to block her from entering the UK, saying: "People who propagate hate speech and division should not be given free entry to the United Kingdom.
"This decision is the right course of action by the UK government and this should be a precedent for others who chose to promote disinformation and hatred."
Last week, Homelessness charity Centrepoint said it had cut ties with Sharon Osbourne after she expressed support for next month's Unite the Kingdom rally.
Police were pelted with eggs and beer cans as people gathered on a residential street in the Surrey town on Monday evening.
In an update last week, officers said there was no evidence so far to support the woman's claim and no evidence that asylum seekers or immigrants were involved.
Surrey Police posted on a local Facebook group on Monday to say the protest had "escalated to public disorder, including missiles being thrown".
"Officers have surrounded a residential property and nearby hotels after they were targeted by protesters purporting that they were locations housing those seeking asylum," the post added.
Police said the information was "wholly inaccurate" and officers had been forced to put on "full protective public order uniform".
They warned anyone committing a crime would face "robust action".
It's the second time police in riot gear have faced off with protesters in the Surrey town since the alleged rape in the early hours of Saturday 11 April.
The woman said she was attacked outside a Methodist church on Ashley Road after leaving the Labyrinth nightclub.
There was some public anger that the force had not released the ethnicity of any suspects but Surrey Police said this was because "the information about the incident and potential suspects is so limited".
"To date, we have not found any evidence of the offence as reported but the investigation is ongoing," said assistant chief constable Sarah Grahame on Friday.
"No descriptions have been released as the information about the incident and potential suspects is so limited," she added.
"To address the specific commentary, there is no evidence that asylum seekers or immigrants were involved."
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The first protest took place last Wednesday when people congregated in Epsom town centre. The group blocked the road before leaving around 8pm.
Danny Tommo, a former associate of right-wing activist Tommy Robinson, had promoted the event online.