After hitting the headlines with a treble at the Cheltenham Festival in 2013, Cooper was handed the coveted role at the start of 2014, replacing Davy Russell.
The combination has gone on to enjoy great success, winning numerous Grade One prizes with the likes of Identity Thief, Don Poli, and, most notably, 2016 Cheltenham Gold Cup hero Don Cossack.
Cooper's latest top-level victory in the maroon and white silks of Ryanair supremo Michael O'Leary's operation was aboard Apple's Jade in the Mares Champion Hurdle at Punchestown in April, also steering her to Cheltenham glory the previous month.
Gigginstown Stud's manager, Eddie O'Leary, said: "There's been no parting company, we'll still be using him a lot, he just won't have automatic first choice as of today.
"We'll still be using and supporting him, but he won't have first call on the horses, that's all."
O'Leary revealed the team did not intend to appoint a direct replacement, adding: "There are a lot of good riders out there.
"Be it Bryan or Sean Flanagan or Davy Russell or Jack Kennedy, we'll be using the best available."
Cooper said in a statement: "I would sincerely like to thank Mr O'Leary and his team at Gigginstown House Stud for the wonderful opportunity of being their retained jockey since January 2014. Their support to me has been immeasurable.
"We have enjoyed great successes and since taking the job I feel privileged and proud to have ridden 28 Grade One winners in the maroon and white silks. Most notably, the 2016 Cheltenham Gold Cup on Don Cossack and other highlighted victories on horses such as Don Poli and Apple's Jade.
"There has been no parting company or falling out and it's business as usual. I look forward to continuing being part of the Gigginstown House Stud team and utilising existing relationships with trainers as a freelance jockey.
"I have no specific targets other than to continue to be successful."
Cooper's agent, Kevin O'Ryan, told At The Races: "Gigginstown are going to use the best available, Bryan being one of those.
"He's had an unbelievable few years with them. He's had plenty of injuries, but he's enjoyed numerous Grade One successes including the pinnacle in the Cheltenham Gold Cup on Don Cossack.
"He's had a lot of success in the Gigginstown colours and hopefully he'll ride many more winners for them. Davy Russell has ridden a lot of winners for them since Bryan Cooper replaced him.
"Henry de Bromhead has his own jockeys, Noel Meade has Sean Flanagan and Jack Kennedy is in Gordon Elliott's. Davy Russell rides a lot for Gordon, as does Bryan.
"Obviously Michael O'Leary feels this is the best policy going forward and you have to respect his wishes."
