October 17, 2012 06:30 PM

Donald Traill/AP

Since giving birth to a baby girl on July 15, Uma Thurman and her beau Arpad Busson have stayed publicly mum on their daughter‘s name. Until now.

And it sure was worth the wait.

“I would like to announce Uma and Arki’s daughter’s name for the first time officially: Rosalind Arusha Arkadina Altalune Florence Thurman-Busson, better known to family and friends as Luna,” the actress’s rep Gabrielle Kachman tells PEOPLE exclusively.

While the couple is keeping the origins of Luna’s name private, Thurman’s rep allows, “Each name has a special reason and meaning to her mother and father.”

Born in Rhinebeck, N.Y., baby Luna is the first child together for Thurman, 42, and Busson, 49. She joins elder siblings, Maya Ray and Levon Thurman-Hawke, from Thurman’s marriage to Ethan Hawke, and Flynn and Cy Busson, the financier’s sons with ex-fiancée Elle MacPherson.

— Elizabeth Leonard

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