Holm, 25, proposed to Morton, 29, on her birthday last Saturday.
"We're over the moon," Morton said in a statement. "We're very much in love and looking forward to spending the rest of our lives together."
Holm -- the son of actor Ian Holm -- directed Morton about a year ago in a music video for The Victims.
The couple is planning a September wedding.
Morton received Oscar nods for "Sweet & Lowdown" in 1999 and "In America" in 2002. The mother of a 6-year-old daughter is getting ready to portray Hollywood bombshell Marilyn Monroe in the upcoming film, "Mr. Lonely."
Celebrity Couples of 2014 [PHOTOS]