The birth was announced on Wednesday morning on Twitter by Abhishek.
Father-in-law Bollywood legend Amitabh Bachchan tweeted he was “overwhelmed” by the congratulatory messages.
The baby – Aishwarya’s first child – was born in the Indian city of Mumbai. Rai gave birth naturally and shunned the common practice of caesarian delivery for over 35 women.
Abhishek’s Twitter page has been inundated with congratulatory messages.
“How beautiful is this feeling. Allah bless you and Aishwarya. Dying to come and see the three of you…” superstar Shah Rukh Khan wrote to Abhishek.
“Congratulations… All the love and happiness… am sure she will be the prettiest girl ever!” tweeted director Karan Johar.
“Been smiling all day at the news. Big yay for this girl and all you amazing Indian women. We need many many more of you,” tweeted fashion photographer Atul Kasbekar.
Amitabh Bachchan wrote on Twitter that he had been responding to messages on the site and also answering individually to more than 1,000 text messages he had received on his cell phone.
There has been tremendous interest in Aishwarya’s pregnancy from the media and her fans ever since it was announced in June by Amitabh Bachchan.
Aishwarya’s pregnancy and the birth of what many are calling India’s “most-awaited baby” had generated a lot of interest but on Wednesday, the television channels were surprisingly restrained in their coverage of the birth.
“As I travel back from the hospital I need to thank the media for keeping their word of not intruding our privacy. Deeply obliged!,” Amitabh Bachchan tweeted late Wednesday.
Often called the “queen of Bollywood”, Aishwarya, 37, married Abhishek in 2007, forging the ultimate Bollywood dynasty.
She first charmed the nation at just 21, winning Miss World in 1994. She has acted in several Hollywood films too.
She was the first Indian actress to sit on the Cannes jury in 2003. She has also appeared on the cover of Time magazine as the global face of Indian cinema.