Tyler and Cameron Winklevoss have decided to drop their lawsuit of Facebook CEO Mark Zuckerberg ending a long legal battle between the two parties.

After vowing to take their dispute to the Supreme Court, lawyers for the brothers Winklevoss said they will not pursue further legal action.

The dispute famously started in 2004 when the Winklevoss brothers along with partner Divya Narendra claimed as undergraduate students at Harvard University, Zuckerberg stole their idea for a social network website. The network, which has since become Facebook, they said was directly a ripoff of their ConnectU idea. The dispute was famously portrayed in the Oscar nominated film The Social Network in 2010.

In 2008, the trio signed an agreement with Zuckerberg which settled the dispute for $65 million. In exchange, the three would drop all litigation against Zuckerberg. However, shortly after the movie came out, they said the agreement was based on inaccurate valuation of the company. Instead, Facebook is worth more than they settled for and they wanted a fair share in the company's value.

The trio took their dispute to the 9th U.S. Circuit Court of Appeals, which agreed with Facebook and Zuckerberg in April and ruled against the Winklevoss. Following this decision against them, the Winklevoss vowed to take their argument to the Supreme Court. However, they have since changed their mind citing undisclosed reasons.

We've considered this case closed for a long time, and we're pleased to see the other party now agrees, Facebook said in a brief statement.

This is the second major lawsuit Facebook has had to deal with concerning its early days. New Yorker Paul Ceglia has tried to argue he made an agreement with Zuckerberg in 2003 that gives him claim to an 84 percent stake in the company.

Follow Gabriel Perna on Twitter at @GabrielSPerna