The former England and Newcastle boss has decided to quit the role following the sacking of Steve Staunton earlier this week.
Sir Bobby Says 'Goodbye'!
Robson was brought in alongside Staunton in January 2006 and will quit his role after next month's game with Wales.
Sir Bobby: "I came in with Steve Staunton and it wouldn't be right to work for another full-time manager.
"I've thoroughly enjoyed my time with Ireland. I am just sorry we didn't qualify for the European Championship and that 'Stan' Staunton paid a heavy price.
"He worked hard to make qualification happen and I think he has the managerial talent to be a success in the future.
"I hope and expect he will be given another chance in the game.
"After talking with 'Stan', I have decided to honour my contract with the FAI and work at the match in Cardiff.
"That decision has Stan's full blessing but after that I will be leaving.
"I'd like to thank chief executive John Delaney and the FAI for all their support towards me over the last couple of years."