At 7:30 Wednesday morning, a DC-9 carrying Vice President Mondale and one aide, Jim Johnson, left Andrews Air Force Base in suburban Maryland on a mission of some delicacy.

Two hours later, they touched down at West Palm Beach Fla., picked up their passenger and, shortly after noon, delivered him, still unobserved, at the White House.

Dr. Arthur F. Burns, 73-year-old chairman of the Federal Reserve Board, walked into the Oval Office and learned, for the first time, from President Carter that he would be replaced in that position by G. William Miller, the chairman of Textron, Inc.

Thus ended one of the most secretive and significant decision-making processes in the first year of the Carter administration, the choice of the person to head the independent and influential central bank.

Although the question of Burns' future had been a matter of speculation and concern in business and political circles for months, White House officials said yesterday that Carter did not begin to focus on it until mid-November.

Burns, a stalwart conservative, had become a symbol of fiscal security to the financial community in his eight years as Fed chairman. He and the President had developed a warm personal relationship in their regular talks during the past 11 months.

But to leading congressional Democrats, including House Speaker Thomas P. (Tip) O'Neill Jr., (D-Mass.) Burns was the living embodiment of tight-fisted, tight-money Republican economics, and they made what one presidential aide called "fairly impassioned pleas" that he not be reappointed when his term expires on Jan. 31.

Shortly before Thanksgiving, Carter asked Mondale and his senior White House staff members to give him their advice on the Burns question. The advice was to make a change, and Carter, without committing himself to accept it, told Mondale in early December to begin screening possible replacements.

The Vice President collected a long list of possibilities from such colleagues as Secretary of the Treasury W. Michael Blumenthal, Charles L. Schultze, chairman of the Council of Economic Advisers, presidential assistants Stuart Eizenstat and Hamilton Jordan, James T. Mclintyre, the acting director of the Office of Management and Budget special trade negotiator Robert S. Strauss and presidential adviser Charles H. Kirbo.

Mondale and the others winnowed the list to seven or eight names - reportedly including two men who when contacted, said they were not available: Reginald H. Jones, chairman of General Electric, and Irving D. Shapiro, chairman of DuPont.

The President had told Mondale that he did not want to drop Burns unless be was convinced he had someone who met his standards to replace him. And those standards were quite specific:

He should be a moderate conservative. Carter did not want to signal a sharp change of direction in fiscal or economic policy.He should be some one recognized and respected in the business-financial community. He need not be a banker or economist as Burns and most of his predecessors had been, but he should have some familiarity with the Federal Reserve System and, importantly, be a strong advocate of its continuing independence.

Further, the President wanted a man with experience in the international field, someone who would be a strong administrator and an effiective spokesman for the central bank.

In addition, there were two very special and almost contradictory tests: If Burns were to be replaced, his successor would have to be a very tough, for the outgoing chairman had the option of remaining on the board for another four years, and Carter could not risk ketting Burns run the Fed on a de facto basis if he had been stripped of his title.

But, for the sake of business confidence and to avoid a nasty confirmation row, it would be very desirable, Cartet indicated, if Mondale and the others could come up with a replacement that Burns would pronounce acceptable.

Interviewing and checks with a range of outside figures - businessmen, economist, academics - cut the list of prospects in half. Two of them came in for visits with Mondale and Jordan on Friday, Dec. 16 - both it happened, men the Vice President had known previously.

One was Bruce K. MacLaury, a 46-year-old moderate Republican, a former president of the Federal Reserve Bank in Minneapolis, and , for the past year, the president of the Brookings Institution.

The other was Miller, the 52-year-old Textron executive, whose service as a governor of the Federal Reserve Bank of Boston and head of an international conglomerate satisfied many of Carter's criteria.

Mondale had known MacLaury as a fellow Minnesotan and Miller as a loyal Democrat and 1968 convention delegate and financial contributor to Mondale's mentor, Sen. Hubert H. Humphrey (D-Minn.).

Both men passed muster, and on Tuesday morning were brought in, separately, for unpublicized interviews with the President.

That afternoon, Carter called in Mondale and other senior officials and told them, for the first time, that he had decided to replace Burns and preferred Miller as his successor.

There remaind only the matter of notifying the two men and arranging a graceful announcement - but that would not be easy. Neither one was in Washington. Miller, vacationing in the Bahamas, was called by Carter and arrangements were made to have him flown to Washington Wednesday.

The matter of breaking the news to Burns worried the White House staff. He had campaigned hard for the job, and the Carter aides felt it would be impolite for him to be notified of his rejection by anyone less than the President, in a face-to-face meeting. Unfortunately, Burns was on holiday in Palm Beach.

So Carter called the chairman Tuesday night, saying that he hated to interrupt his vacation but hoped he could come to the White House on Wednesday and would send a plane to pick him up. He did not tell Burns what he had decided.

Then carter and Mondale hit on the idea of the Vice President flying down to get Burns, as a further gesture of respect. But Mondale was under the same rule of silence on tipping Carter's hand, so he and Burns passed the time discussing the condition of the dollar, the human rights campaign, volunteerism and other relatively safe topics.

When Burns reached the White House, he and Carter spent an hour together and reportedly, despite his disappointment. Burns took the decision not to retain him in the chairmanship well. Then, he, Carter, Mondale and Miller went out to face reporters and make the decision public, with Burns declaring that the President has chosen "wisely and well."

Wednesday evening, the outgoing and incoming chairmen of the Fed shared the same White House plane on their return trip with Burns dropping off in Palm Beach and Miller continuing on to his vacation in the Bahamas.

Carter and Mondale breathed large sighs of relief.