The 29-year-old model and her fiance Wiz Khalifa welcomed their 'perfect' first child Sebastian Taylor Thomaz into the world on Thursday.
Wiz, 25, tweeted: 'Happy Birthday Sebastian "The Bash" Taylor Thomaz! Everyone welcome this perfect young man into the world.'
Amber then added her own message: 'Happy Birthday to our baby boy Sebastian "The Bash" Taylor Thomaz! I'm so happy u guys I fell in love all over again! :-)'
Stars were quick to add their congratulations.
Rihanna, who herself turns 25 today, wrote: 'Congratulations to my fam @mistercap and @muvarosebud !!! Welcome baby Bash! You're born in an exquisite month #babypisces.'
And it seems that February 21st is an auspicious day for rapper birthdays, with Snoop Dogg also tweeting that it is also his sons.
'Welcome 2 fatherhood @mistercap!!! Congrats 2 u n @muvarosebud on ur new son - he got the same Cday as @cbroadus!!! Bless up.'
Rapper Wiz had earlier posted a picture of himself in doctor's scrubs, including a shower cap and a face mask.
Appearing to ready himself for his pregnant fiancee to give birth, he captioned the photo: 'Daddy time.'
Amber hinted she was close to her due date last week when she announced that she would not be online again until after their baby's arrival, as she wanted to focus on relaxing.
Writing on her Twitter page, she said: 'Hey Rosebuds & Rosestuds Muva won't be back on twitter until Wizbud is born I need 2 meditate & focus on our Lil boy Luv u talk soon - Muva (sic)."
She told Los Angeles radio show Big Boy's Neigbourhood earlier this month: 'I don't want no medication. We take home birthing classes to teach us how to do it, and we're gonna have our midwife, and when the baby's head comes out, he's gonna be Dr. Wiz.'
'He's gonna be the first person to touch the child. No doctors, no gloves,' added the model.
There was speculation that Amber went into labour earlier this week after Wiz posted an Instagram picture of a fetal monitor, saying 'Coming soon!'.
But the model, who previously dated Kanye West, set the record straight on Wednesday, tweeting: 'Good morning Rosebuds & Rosestuds Muva is not in labor nor did I have the baby yet I just had a doc appt yesterday Lol that's all. ;-)'
After the couple's child is born, Amber and Wiz will resume their wedding plans.
'It makes more sense for the baby to be in the wedding than for her to be pregnant,' said Wiz. 'Some people, they rush and do it, but it works both ways.'
Amber added: 'Being pregnant, I'm tired a lot. So to have a wedding and planning it all out and the reception - I just don't want to be sleeping when everybody's partying at our wedding.'