"Both mom and baby are doing brilliantly," says the British actress's rep

British actress Samantha Morton and her fiancé Brit Harry Holm have welcomed a baby girl, PEOPLE has learned exclusively.

The couple’s daughter, who has not yet been named, was born in an undisclosed London hospital early Jan. 4.

“It is fantastic news and both mom and baby are doing brilliantly,” Morton’s publicist, Karon Maskill, tells PEOPLE.

Under English law the couple has six weeks in which to register the child’s name.

Morton, 30, met Holm, 27, in 2005 when he directed her in a video for the band The Victims. Double Oscar-nominee Morton (for 1999’s Sweet & Lowdown and 2002’s In America) already has a daughter, Esme, from her relationship with British actor Charlie Creed-Miles.

Holm proposed to Morton in Paris in May, 2006 – on her 29th birthday. The pair has put back their wedding day due to career and now personal commitments.

PEOPLE revealed exclusively the actress’s pregnancy last August.