It is the seventh child for Alec, who also has 25-year-old daughter Ireland.
A representative for the couple confirmed the news to People magazine, adding that no further statement will be made, beyond Hilaria's social media post.
The couple has not addressed speculation over whether they used a surrogate or adopted the newborn girl.
Social media fans have also pointed out the fact that, back in November, Hilaria said in an interview she wasn't sure whether they would expand the family - going as far as calling the idea "insanity".
"During times of COVID, it very much feels like we're done," she said just over three months ago.
"Everybody's asking me this question. I don't know. I've said in the past that I was done when I wasn't done. I think, right now, I'm so tired," she added. "And I feel, just with Covid – it's just insanity," she added.
The Baldwin family was at the centre of a scandal in recent months after Hilaria was accused of grossly exaggerating her Spanish heritage.
A social media user uncovered what they described as her "decade long grift", pointing out the multiple instances where Hilaria claimed to have been born in Spain.
Earlier this month, Hilaria issued an apology on Instagram.
"I've spent the last month listening, reflecting, and asking myself how I can learn and grow. My parents raised my brother and me with two cultures, American and Spanish, and I feel a true sense of belonging to both," she wrote in the post.
