Controversy erupted last week after activist investor Daniel Loeb of hedge fund Third Point accused Thompson of adding a bogus bachelors computer science degree to his CV.
In the memo, seen by Reuters, Thompson said: "I want you to know how deeply I regret how this issue has affected the company and all of you.
"We have all been working very hard to move the company forward and this has had the opposite effect. For that, I take full responsibility, and I want to apologize to you."
Thompson said he would "respect" the board's plans to conduct a thorough and independent review and hoped it would be "concluded promptly".
According to reports, Third Point, which owns 5.8% of Yahoo shares has issued a legal demand to review how much the Yahoo board knew about Thomspon's qualifications before it hired him as chief executive in January.
He assured Yahoo employees that he remains focused on taking the steps that he believes are needed to revive Yahoo's revenue growth and boost its long-sagging stock, even with his own job imperilled.
The episode is a major blow for Thompson, with many commentators speculating that this could cost him his job.
His short stint at the company has been eventful, first with Yahoo's move to sue Facebook over alleged patent infringement, shortly followed by a company wide restructure which will cut 2,000 jobs.
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