Williams has an opt-out in his contract after the fourth year, sources told ESPN.com's Marc Stein.
"Officially a Brooklyn Net! Signed my contract on an iPad Just thank God for this wonderful Opportunity #HelloBrooklyn" Williams tweeted.
"We are obviously thrilled that Deron will lead the Nets into our new home in Brooklyn," Nets general manager Billy King said in a statement. "Deron is in my opinion the top point guard in the NBA. With his leadership and guidance on and off the court, the fans of Brooklyn should be proud."
Williams announced on Twitter last Tuesday night that he had decided to stay with the Nets over leaving to play in his hometown with the Dallas Mavericks.
"I am really excited to be able to officially say that Brooklyn is home," Williams said in a statement on Wednesday. "It is an incredible time to play in Brooklyn and to be a part of this organization, the first sports team to call Brooklyn home in over 50 years."
"I actually thought that's where I was going to go," Williams said. "I had the meetings and it kind of changed my mind because once I got out of the meeting with Dallas and saw the way they were going and the team they were putting out there, and I saw that we just made a trade for Joe Johnson and I felt like that team for a longer time would be the better team."
Williams is in Vegas for Team USA's training camp prior to the Olympics.
The Nets have not made the Williams signing, the Johnson trade or their bevy of other moves -- re-signing Gerald Wallace, signing Mirza Teletovic and acquiring Reggie Evans in a sign-and-trade deal -- official yet, but are expected to do so soon.
The Nets have also been engaged in blockbuster trade talks with the Orlando Magic about Dwight Howard, but the Los Angeles Lakers and Houston Rockets have also entered the mix as possibilities to land the 26-year-old center, sources told ESPN.com's Marc Stein.
At the same time, the Nets have been engaged with restricted free agent center Brook Lopez's representatives about a contract extension. Lopez would be the centerpiece in a deal for Howard.
"There are a lot of balls up in the air right now," a league source told ESPN NewYork.com.
Mike Mazzeo is a regular contributor to ESPNNewYork.com.