The broadcaster, 36, said it was “time to go and maybe make some other dreams come true” as he told listeners he would be leaving the station after 14 years.
Grimshaw told the station it is an “exciting” and “happy” step to take.
“It has been my home for 14 years, it was the only place I ever wanted to work,” he said.
“I just wanted to say how I have loved it and how, honestly, I have had the time of my life.”
In an official statement, Grimshaw said working at Radio 1 had been a “childhood dream”.
He added: “I have been lucky enough to make that dream come true. It has been everything I’d imagined and even more.
“I grew up wanting to connect with people and to feel accepted and the Radio 1 listeners gave me that and let me be part of their daily life, for which I will be eternally grateful.
“But over the last few months I’ve been doing a lot of thinking about my future and, after 14 years, I’ve made the decision that it’s time for me to move on – I’d like to thank the listeners, as without them none of this could have been possible, and the Radio 1 family, who have been such a huge part of my life.
“Thank you, thank you, thank you.”
Hope said taking over Grimshaw’s show was “madness”.
She added: “I love Radio 1 so, so much, having grown up tuning in religiously every single day, it means the world to be taking the reins at home-time with the phenomenal Mr North.
“Grimmy is an absolute legend of the game, thank you mate for every laugh you’ve sent rippling through the country!
“Taking over drive is a task we won’t be taking lightly, but Jordan and I are ready to put our all into making your journey home that little bit brighter.”
North, who took part in last year’s series of I’m A Celebrity… Get Me Out Of Here! said: “I am absolutely chuffed to be making the move over to Radio 1 daytime and even happier to be working alongside Vick.
“Grimmy is a Radio 1 legend so we definitely have big shoes to fill but will work extremely hard to make sure afternoons on Radio 1 still sound superb. I am super excited to get started and look forward to this next chapter with Vick and the rest of my Radio 1 family.”
Grimshaw joined the BBC in 2007, hosting BBC Two youth music show Sound alongside DJ Annie Mac.
He went on to host Radio 1’s Weekend Breakfast Show in 2008 until the following year, before moving to a late-night slot.
Grimshaw, who was raised in Royton, Oldham, took over Chris Moyles’ breakfast show at the age of 27 in 2012.
In 2015, he appeared as an X Factor judge alongside Simon Cowell, Rita Ora and Cheryl, but departed after a year.
In May 2018, Grimshaw became the second longest-running host of the Radio 1 breakfast show, overtaking Tony Blackburn.
He then moved on to present the drivetime programme.