Former Miss World Rai Bachchan, 38, gave birth at the private Seven Hills Hospital in Mumbai, hospital staff told AFP.
“IT’S A GIRL!,” Abhishek, 35, wrote on his Twitter account, @juniorbachchan, without giving any further details.
Bachchan’s father, Bollywood legend Amitabh, tweeted his joy at the new arrival: “A dada ji (a grandfather)…Ecstatic!”, calling his new granddaughter “the cutest baby girl”.
Rai Bachchan and her husband are Hindi-language cinema A-listers and — with Amitabh and his actress wife Jaya — are considered to be Bollywood’s “first family”.
Amitabh, the family patriarch, is already a grandfather as his daughter, Shweta Nanda, has two children.
Bachchan senior — a compulsive blogger and tweeter of his thoughts on life — has had a prickly relationship with the media in the past and has criticised outlets that misrepresent him or intrude on his family’s privacy.
Before the birth, Indian broadcasters agreed on an unprecedented series of measures, including not to trail the event in advance or even run the announcement on “breaking news” captions.
Outside broadcast vans would not be sent to his home or the hospital and no astrology shows, which are popular in India to forecast future fortunes, would be aired.
Stories on the subject would also be restricted to between one minute and ninety seconds.
Indian television news channels on Wednesday appeared to be upholding the agreement, with muted coverage.
Aishwarya Rai Bachchan took the Miss World crown in 1994 and made her debut in Bollywood in the late 1990s. She has since gone on to star in more than 40 Indian films.
She was most recently seen in the South Indian film blockbuster “Endhiran” (Robot) with the Tamil-language superstar Rajinikanth, which broke box office records in India.
She has also appeared as a jewel thief in Steve Martin’s “The Pink Panther 2”. Her other English-language films include “Bride and Prejudice” (2004), by “Bend It Like Beckham” director Gurinder Chadha, and “The Last Legion” (2007).
Before her pregnancy was announced, Rai Bachchan was at this year’s Cannes film festival to promote her latest project “Heroine”.
Filming had been due to start this year but in June its producers said the movie had been put on hold because of the pregnancy.