The Premier League club have beaten a large number of clubs including Manchester City to secure the player, with Gilmour having spent the past week at Cobham Training Centre finalising legal arrangements.
Goal intiially reported the deal back in March, with the Scottish Premiership side having now officially confirmed the transfer.
Rangers were ready to fast track Gilmour into their first team to keep him at the club but he was well known across Europe and had been attracting scouts from Arsenal, Manchester United, Bayern Munich and Barcelona.
Gilmour is currently too young to move but he will be able to do so from his next birthday, which falls on June 11.
The midfielder put in a string of incredible performances for his club at Under-20 level, while his showings for Scotland in the Victory Shield also caught the eye of scouts.
The Ayrshire schoolboy was expected to become Rangers' youngest ever player after training with Pedro Caixinha's squad, but his pathway into the first team will now likely be affected by the agreement with the Premier League leaders.
Confirming that Gilmour would move to Chelsea, a statement on Rangers' official website read: "Rangers can confirm that agreement has been reached for our under-16 player Billy Gilmour to join Chelsea in the summer.
"The club’s preference would have been for Billy to stay at Rangers but when he made clear his desire to move to the Premier League it was important that we maximised the commercial value for him. We believe we have done this.
"The agreement will provide Rangers with a significant sum for a young player who has yet to play first-team football, and further significant amounts could be due based on milestones being reached. We are pleased with this outcome and believe this deal represents good value for Rangers.
"That he was sought after by many Premiership clubs, and indeed a number of the largest clubs in Europe, further demonstrates the quality of player now being produced by the Academy. Everyone at Rangers wishes Billy well for his future."