President Carter, his eyes brimming with tears, yesterday accepted the resignation of his close friend Bert Lance as the director of the Office of Management and Budget.

"I accept Bert's resignation with the greatest sense of regret and sorrow," the President told a nationally televised news conference. "He is a good man . . . He is close to me and always will be, and I think he has made the right decision because it would be difficult for him to devote full time to his responsibilities in the future."

The resignation, the result of weeks of allegations about Lance's free-wheeling financial style and conduct as the head of two Georgia banks before he joined the government, brought to an abrupt halt the public career of Carter's first Cabinet-level appointee, a man whom the President described yesterday as irreplaceable to him.

But in accepting Lance's rsignation, Carter was as adamant as ever in defending his friend's competence and itergrity, employing language similar to that of his news conference one month and three days ago when he declared, "Bert, I'm proud of you."

"Nothing that I have heard or read has shaken my belief in Bert's ability or his integrity," the President said yesterday.

It was clear, both from Carter's remarks and Lance's letter of resignation, that the controversy surrounding him would not disappear and would eventually inflict severe damage on the entire administration.

The President acknowledged that the Lance affair had diverted his attention and taxed Lance's time to deal with government matters. He said Lance had personal reasons for wanting to return to Georgia, particularly the need to deal with his mounting personal financial problems.

Morever, Carter added, all the publicity surrounding the controversy had created public doubts "about the integrity of me and our government."

In his letter to the President, Lance said he hoped the American people would feel that "I have met well my responsibilities and performed well my tasks." He said he was convinced he could continue as an effective budget director.

"However," Lance wrote, "I have to ask the question, at what price do I remain?"

Implicitly answering that the price was too high. Lance said he was resigning "because of the amount of controversy and the continuing nature of it."

It was unclear last night just when Lance will actually leave the OMB and return to Georgia. His deputy, James McIntyre, 36, who served in the Georgia state government, will replace him at least temporarily. Carter said he has given no thought to a permanent successor in one of the most powerful posts in the federal government.

Lance departure from Washington, however, will not end a variety of official inquiries into his conduct as the head of the Calhoun, Ga., First National Bank and the National Bank of Georgia in Atlanta. These include investigations by the Justice Department, the Securities and Exchange Commission, the Internal Revenue Service and the comptroller of the currency.

Yesterday's news conference was the President's most emotional public moment since his return to his home town of Plains. Ga., the day following his election victory. At the beginning of it, he paused several times, apparently to gain control of his emotions. But by the end, Carter was smiling often and seemed totally self-assured.

The resignation climaxed weeks of speculation and growing pressure in recent days from members of Congress, including Senate Majority Leader Robert -. Boyd (D-W, Va.), for Lance to leave government service.

It was, Carter and his aides insisted, Lance's decision alone, made without any pressure from the White House.

Lance informed the President of his decision early Tuesday night as the two men sat alone beside the White House tennis court. They had just finished a doubles match against Carter's political adviser, Hamilton Jordon, and his chief speech writer. James Fallows. They talked for about 45 minutes.

Early yesterday, before 7 a.m., Lance met alone with Carter at the White House. He returned home for lunch with his wife. LaBelle, and then the two of them drove downtown. Hand-in-hand, they strolled across the driveway that separates the White House from the Executive Office Building to meet one last time with the President.

Lance left his office in the Executive Office Building at 4:30 yesterday afternoon and was at his Georgetown home when his resignation was announced.

The Lance affair, which so preoccupied Washington during the late summer and has been in White House press secretary Jody Powell's words, the most "personally difficult" crisis of Carter's presidency, was set off by a presidential request to the Senate Governmental Affairs Committee.

In taking over the OMB post, Lance had pledged to sell his substantial stock holdings in the National Bank of Georgia by the end of this year. But the stock dropped in value and Lance faced the prospect of accepting a heavy personal financial loss. Carter asked the committee to extend the deadline for selling the stock.

That set off new inquiries into Lance's banking background and a torrent of new stories about questionable practices such as huge overdrafts on Lance's personal bank account and those of his wife and relatives.The comptroller of the currency issued three reports on the subject, and while White House officials claimed each exonerated Lance, the reports also raised new questions and further tarnished his image as the nation's budget director.

The climax of these inquiries came last week when Lance, in three days of televised hearings, strongly detended his reputation before the Governmental Affairs Committee, often making the committee members look foolish. But while Carter aides were cheered by Lance's performance before the committee, they conceeded in private that he remained a wounded member of the administration.

The controversy was made all the more difficult for the President by his closeness to Lance, whose outgoing nature and backslapping style made him one of the most personally popular figures in the administration.

Carter did not shy away from the subject of the relationship yestersday.

"I don't think there is any way that I could find anyone to replace Bert Lance that would in my judgement be as competent, as strong, as decnent and as close to me as a friend and adviser s he has been," he said.

" . . . There has been a special relationship between me and Bert Lance that transcended official responsibilities or duties or even governmental service over the last six or seven years. So he has occupied a special place in my governmental career, in my political career and in my personal life. I don't think there is any way anyone could replace him now."

After reading Lance's letter of resignation. Carter opened his news conference with a declaration that made clear the personally agonizing aspects of the day's events.

"Bert Lance is my friend," he said. "I know him personally, as well as if he was my own brother. I know him without any doubt in my mind or heart to be a good and an honorable man."

But the President added that he may have been responsible for much of Lance's troubles "because of the extraordinary standards that we have tried to set in government" and because his campaign raised expectations of a higher than normal level of morality in the new administration.

In the light of all the circumstances, Carter characterized Lance's decision to resign as "a courageous and also a patriotic gesture on Bert's part."

The President conceded that he probably has been damaged by the weeks of controversy and his stout defense of his friend. But he indicated that he stuck with Lance through the furor so that the budget director would have a change to defend himself publicly at the Senate hearings.

Carter said he could not say whether, had Lance decided to remain in office, he would have sought his resignation.

" . . . I was honest with him." the President said. "I didn't artificially try to talk him out of it because as we discussed the same facts and the same issues and the same prospects for the future, I think that our minds were working in the same direction."

Yesterday's news conference, orginally scheduled for 3 p.m., was postpned until 5 p.m. because Lance wanted to talk to his attorney, Clark M. Clifford, and have Clifford help him draft his letter of resignation. Clifford returned to Washington from Detroit yesterday afternoon.

The Lance affair was the only subject Carter was questioned about and at times he seemed impatient, characterizing some of the questions as repetitive. The President ended the 35-minute news conference himself without waiting for the customary "thank you" from the senior wire service reporter in the room.