The board of the social media company agreed to a takeover offer from Musk.
Less than two weeks ago, Mr Musk made a shock bid on the social networking site.
He called for a series of changes to be made to eradicate fake accounts.
The firm will now ask shareholders to approve the deal after initially rejecting it.
According to Forbes magazine, Mr Musk is the richest man in the world, with an estimated net worth of $273.6 billion. He is also the CEO of the firm.
Mr Musk said in a statement that free speech is the bedrock of a functioning democracy and that the digital town square is where matters vital to the future of humanity are debated.
I want to make the product better than ever by enhancing it with new features, making it open source to increase trust, and authenticating all humans, he said.
I look forward to working with the company and the community of users to unlock it.
The chair of the board said that Mr Musk's offer was the best path forward for the company's stockholders.
The deal will see the delisting of the shares of the company and it will be taken private.
Mr Musk thinks this will allow him to make changes to the business.