The former Girls Aloud star, 35, and One Direction member, 24, confirmed the decision in posts on social media.
The couple have a son, Bear, who was born in March last year.
They said they were "sad" to make the announcement, and it had been a "tough decision to make", adding: "We still have so much love for each other as a family".
Cheryl tweeted: "Bear is our world and we ask that you respect his privacy as we navigate our way through this together."
Newcastle-born Cheryl started her career on ITV's Popstars: The Rivals in 2002 as one of the members of Girls Aloud. She then went on to launch a solo career and became a judge on The X Factor in 2008.
In 2016, the twice-married star, who was born Cheryl Tweedy, said she wanted to be known just by her first name.
Her relationship with Payne began after her divorce from French restaurateur Jean-Bernard Fernandez-Versini. She was previously married to the former England footballer Ashley Cole.
Payne, originally from Wolverhampton, has gone on to forge a successful solo career after One Direction decided to take a break in 2015, with his song Strip That Down being streamed globally more than a billion times.
In March, Payne admitted he and Cheryl had relationship "struggles", amid tabloid press reports about their life together.