He had earlier said the baby was “a few hours away”. Musk has since replied to a Twitter user and said Grimes gave birth to a boy.
Grimes, 32, has been keeping fans updated throughout her pregnancy and in February revealed she had suffered “complications” and felt she was “woefully ill-prepared” to have a baby.
Later that month she told fans on Twitter she will allow the child to choose its own gender.
Musk and Grimes have been dating for more than two years and made their red carpet debut as a couple at the 2018 Met Gala.
South African-born Musk, one of the world’s most powerful and richest men, has five children with his first wife, the Canadian author Justine Wilson.
He has been in the headlines recently for his criticism of the coronavirus lockdown in the US.
He tweeted: “FREE AMERICA NOW.”
Elon Musk and Grimes welcome their first child together
