Gilmour has been in and around the Chelsea senior team this season, making five appearances in all competitions, including two in the Premier League earlier in the campaign.
The 18-year-old Scotland youth international has made most of his appearances for Chelsea’s u-23 side.
Gilmour is the latest member from Chelsea’s academy to break through under Lampard this season, joining the likes of Tammy Abraham, Reece James, Fikayo Tomori and Mason Mount.
Gilmour faces competition for places in a Chelsea midfield that includes the likes of Jorginho, N’golo Kante, Mateo Kovacic, Mount, Ross Barkley and Ruben Loftus-Cheek.
"It's a massive moment for me and I'm really happy," Gilmour told Chelsea's official website.
"I went with the squad to Leicester and, even though I wasn't involved, the gaffer [Lampard] pulled me afterwards to tell me that after the winter break I'd be with the first-team permanently.
"It's a dream come true for me. It's the first big step I think, moving into the first-team dressing room at a club like Chelsea. It's massive for me.
"Knowing that the manager has such faith in you is massive, it gives me so much confidence when I step onto the field. It's really good knowing that he trusts in you and likes the way you play, and I think that's the same for everyone here.
"Age isn't really an issue here. The gaffer has shown that if you work hard and perform well, then you will get opportunities. Hopefully I can keep pushing on and work hard every day in training."
