Mr Musk has suggested this will give him freedom to make the changes he wants to the business.
Twitter shares on Monday closed more than 5% higher after the deal was announced.
By buying Twitter, does Elon Musk have a political project in mind?
In practice Twitter + Musk will mean a much lighter moderation policy.
Others have been left dismayed at what Twitter might look like without strong platform moderation.
Musk was invited to join Twitter's board but turned down the offer before launching a surprise bid for the company on 14 April, saying he wanted to "unlock" its potential as a bastion of freedom of speech.
News of the takeover has been cheered by the right in the US, although Mr Trump on Monday told Fox News he had no plans to re-join the platform.
Twitter is currently led by Parag Agrawal, who took over from co-founder and former boss Jack Dorsey last November.