The Lord Of the Rings actress, 37, announced the rather unusual name via an Instagram snap on Friday (20 February).
Sharing a glimpse of the baby boy - albeit only his tiny hand! - Liv simply told fans: "Sailor Gene Gardner."
The caption was accompanied by heart emoticons as she expressed her love for her newest family member.
In the picture, Sailor can be seen with a anchor pendant in his hand.
Liv gave birth on 11 February, and a spokesperson confirmed the baby was "happy and healthy".
Sailor is Liv's first child with partner Dave Gardner.
She also has a 10-year-old son Milo with ex-husband Royston Langdon. Sports agent Dave - who is best friends with David Beckham - has seven-year-old son, Grey, with British ex-wife Davinia Taylor.
The very private couple are believed to have met in early 2014 via mutual friend Kate Moss.
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