The actress, who will get a £5.6million divorce payout, has told friends she is “smitten” with Elon – the boss of electric car giant Tesla – and they are now making plans together.
It comes as it emerged the 30-year-old will get to keep the dogs, Pistol and Boo, she shared with Pirates of the Caribbean star Johnny, 53.
A source said: “Amber is over the moon about her divorce but she is even happier about being able to go public with Elon.
“She was telling friends over Christmas how smitten she is with Elon.
“They are already planning a lot of adventures together and she’s relieved she can move on with her life.It’s an exciting new chapter for both of them.”
Billionaire Elon, 45, had been reportedly trying to court Amber for four years before the pair were spotted out together in London last August.
They both appeared in cult director Robert Rodriguez’s 2013 film Machete Kills, but never met on the set.
Elon divorced his British wife Talulah Riley for the second time in November.
They were married between 2010 and 2012 and again from 2013 to 2016.
Elon is worth an estimated £9.2billion and the South Africa-born magnate also owns SpaceX – a California-based space exploration company.
He reached an undisclosed settlement with his now estranged wife.
Elon and actress Amber grew close as she endured a difficult break-up.
She had accused Johnny of domestic abuse – a claim he denies.
The couple, who have no children together, had been married for less than 18 months when they split in May last year.