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When Michael S. Dell’s takeover bid for the company he founded appeared to be in peril on Wednesday, he sprang into action to try to save it.

After a special board committee rebuffed his request to change the voting rules for his nearly $25 billion offer, Mr. Dell held talks with his main partner in the deal, Egon Durban of the investment firm Silver Lake Partners, about a potential compromise, a person briefed on the matter said. The men, who were working from their homes in Hawaii, reached out to the special committee late Wednesday.

By Thursday evening, the two sides began completing a revised agreement and announced new terms Friday morning, keeping the takeover effort alive for at least another month.

Under the terms of the new offer, Mr. Dell and Silver Lake will pay $13.75 a share plus a special dividend of 13 cents a share. Shareholders would still receive a regularly scheduled third-quarter dividend of 8 cents a share.

Related Links Document: The Dell special committee statement

Mr. Dell is effectively financing the special dividend by taking a bigger discount on the nearly 16 percent stake that he is contributing toward the buyout, people briefed on the matter said. He is now valuing his shares at a little more than $12.50 each, down from $13.36.

In return, the special committee agreed to change the rules by no longer counting Dell shares not cast in a special election as “no” votes. The current rules count absentee votes as “no” votes, creating a high hurdle. (Mr. Dell is still not allowed to vote his shares in the deal.)

A shareholder vote on the original bid of $13.65 a share, which had been scheduled for 10 a.m. Eastern on Friday, was adjourned a third time, to Sept. 12.

Perhaps most important, the record date, or the day by which investors must have held Dell shares to be eligible to vote, was moved to Aug. 13 from June 3. That will allow a number of investors favorably disposed toward the deal, like arbitrageurs who bet on the outcome of mergers, to participate in the voting.

The new changes are meant to lock up a deal that has faced numerous challenges, notably by the longtime shareholder activist Carl C. Icahn. Had Friday’s vote proceeded as planned, it would most likely have led to the bid’s defeat.

Instead, both the Dell special committee and the prospective buyers claimed victory for themselves and for shareholders. A number of big investors leery of the original buyout bid, including the mutual fund manager Franklin Mutual Advisers and the hedge fund Pentwater Capital, indicated on Friday that they would support the new offer.

Shares in Dell jumped 5.6 percent on Friday, to $13.68, reaching levels unseen since April.

The chairman of the special committee, Alex Mandl, said in a statement: “The committee is pleased to have negotiated this transaction, which provides as much as $470 million of increased value, including the next quarterly dividend that will now be paid regardless of when the transaction closes.”

The compromise also reduces the size of the breakup fee, to $180 million from $450 million, that would be paid if the buyout deal is terminated and Dell then undertakes a recapitalization transaction “that does not result in there being an absolute majority stockholder of the company.”

Mr. Dell and Silver Lake are privately trumpeting that they have changed what they call an unfair voting standard. They have argued that it is impossible to read the intentions of all shareholders who have abstained from voting because some are investors who do not vote or do not even know that they still own shares. Others, like Mr. Icahn, have contended that a significant number of absentee votes remain opposed to Mr. Dell’s takeover approach.

Last week, Mr. Dell and Silver Lake first proposed raising their takeover bid to $13.75 a share in exchange for changing the voting standard. But the Dell special committee balked at making what it regarded as a significant concession without winning more money for shareholders, people briefed on the matter said previously. The directors had been hoping for a bump to at least $14 a share.

Mr. Icahn, who sued Dell in Delaware’s Court of Chancery on Thursday to prevent such changes to the deal, wrote in a tweet on Friday: “We are pleased to have won another battle in the Dell war, but the war itself is far from over. More to follow.”

Southeastern Asset Management, which has been an ally to Mr. Icahn, said in a statement, “Stockholders should ask why the special committee is acting as though its mandate is to get this deal done at any cost necessary when the transaction is so stockholder-unfriendly that it could not receive the required stockholder approval on three occasions.”

Mr. Icahn has objected to the Dell board’s delaying of a vote on the deal, which in turn has pushed the annual meeting back to October. Mr. Icahn, who is seeking to replace all 11 directors with his own slate, has argued that the delays run counter to Delaware law, since the annual meeting will be held later than 13 months after the last one.

But Lawrence A. Hamermesh, a professor at the Widener University Law School, said that Delaware rules did not require that annual meetings be held within that time period. Instead, they give shareholders the right to demand such a meeting after 13 months have elapsed — and it would still take the company weeks to organize a gathering even after a court order.