A £500,000 fee has been agreed for the player, who trained with the Rangers first-team earlier this year.
The Ibrox club have tried in recent weeks to convince the highly-rated academy prospect that his future belonged in Glasgow.
However, Gilmour had his heart set on a move to the English Premier League and will officially become a professional at Chelsea when he turns 16 in June.
A statement on the Rangers website read: “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.”