Amitabh (L), daughter-in-law Aishwarya and son Abhishek are often called Bollywood's 'first family' [EPA]

Indians are making a song and dance after Bollywood's "first family" announced the arrival of a baby girl despite tight restrictions on media coverage.

Aishwarya Rai Bachchan, an actress and former miss world, gave birth to the baby on Wednesday morning, according to a tweet by Abhishek Bachchan, her actor husband.

Abhishek is the son of Bollywood legend Amitabh Bachchan, India's most famous actor whose film career spans more than four decades.

"Its a girl!" Abhishek tweeted after the birth at the private Seven Hills  Hospital in Mumbai, while Amitabh tweeted: "A dada ji (a grandfather)...Ecstatic!", calling his new granddaughter "the cutest baby girl".

Amitabh, the family patriarch, is already a grandfather as his daughter, Shweta Nanda, has two children.

Aishwarya's pregnancy generated tremendous interest in the Indian media and among fans with bookmakers taking an estimated 1.5 billion rupees (around $30m) in bets on whether she would deliver on 11 November or not.

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.

Indian television news channels on Wednesday appeared to be upholding the agreement, with muted coverage.

Instead, it was left to the Internet for fans to wish the Bachchans well as congratulations poured in via Twitter, with #BabyB quickly trending among Indian users of the micro-blogging site.

"Bachchans blessed with a Girl!" said India's NDTV television channel, urging users of its website to "Send your wishes for Baby B."

Aishwarya won 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.

Often called the "queen of Bollywood", Aishwarya married Abhishek in 2007, forging the ultimate Bollywood dynasty and even appearing on the cover of Time magazine as the global face of Indian cinema.