“We are excited to partner with Twitter as an investor and a member of the board,” Mr. Durban said in a statement. “Jack is a visionary leader, and a critical force behind Twitter’s ongoing evolution and growth.”

But the settlement also lays the groundwork for eventually replacing Mr. Dorsey. Twitter agreed to create a new committee of five directors, including Mr. Durban and Mr. Cohn, that would study the company’s succession planning and leadership structure and recommend any changes by the end of the year.

Companies that Elliott has set its sights on have ended up replacing their chief executives after the hedge fund gained a board seat. EBay’s previous chief executive, Devin Wenig, resigned months after Mr. Cohn became a director of the e-commerce company, for example.

“We invested in Twitter because we see a significant opportunity for value creation at the company,” Mr. Cohn said in the statement announcing the settlement. “I am looking forward to working with Jack and the board to help contribute to realizing Twitter’s full potential.”

Elliott’s campaign was not the first leadership challenge that Mr. Dorsey had faced in his tenure at Twitter. In 2008, he was fired from the social media company he co-founded, then was brought back in 2015. A year later, Mr. Dorsey withstood an acquisition effort by Salesforce.

Twitter’s share price has not increased significantly since Mr. Dorsey’s return in 2015, when it was $36. At the close of trading on Monday, shares were $32.46, down 3 percent for the day.

Analysts said investors would welcome the news, although it was overshadowed by plunging markets driven by fears of the coronavirus spread. “The most important thing from an investor standpoint: Silver Lake is making a billion-dollar investment in Twitter at a time when the market is in complete panic mode,” said Richard Greenfield, a partner at LightShed, a technology and media research firm.