(PEOPLE.com) -- Phoenix frontman Thomas Mars is quickly becoming outnumbered at home he and director Sofia Coppola recently welcomed a second baby girl, a source confirms to PEOPLE.

Daughter Cosima was born in New York City "within the last month" and joins big sister Romy, 3.

The couple, who began dating in 2005, announced the pregnancy in December.

Mars continues to tour in support of his band's Grammy-winning album, "Wolfgang Amadeus Phoenix," while Coppola's latest film, "Somewhere," will be released in December.