According to Spain's Hola! magazine, the pair welcomed a son in NY "a few days ago", with further details about the tot's name and birth weight yet to be revealed.
Screen icon-turned humanitarian activist Gere has a son aged 18, Homer James Jigme, with former wife Carey Lowell. The 35-year-old publicist posted a since-deleted photo of her growing baby bump getting blessed by the Dalai Lama! "Very happy receiving blessings for our precious treasure to arrive..."
Alejandra, who was born in Spain's north-west region of Galicia, is already mother to a six-year-old son called Albert with her ex-husband Govind Freidland.
Silva and Gere married previous year in a secret ceremony. It's his third marriage; he was previously married to Cindy Crawford from 1991 to 1995, and Lowell from 2012 to 2016.
After news of their marriage broke, a film source who has known Gere for years told PEOPLE he "has been much happier in this half of his life because he is settled, confident and doing what he likes". "I'm married to a lovely woman who is smart, sensitive, (and) committed to helping people, who's fun, patient, who knows how to forgive, who's a great cook - and who makes the best salads in the world!"
The pair met in 2014 when Gere was staying in a hotel owned by Silva's family.