The Federal Reserve Board yesterday unanimously approved another sharp rise in interest rates, a step it hopes will stem inflation and stop the slide of the dollar, but that also could worsen the recession.

In its first formal action under the new chairman, Paul A. Volcker, the seven-member board voted to raise its key discount rate from 10 to a record 10.5 percent -- the second time in less than a month that it has acted to tighten credit.

The move is likely to pose new difficulties for the Carter administration, which recognizes inflation as the nation's No. 1 economic problem but is struggling to avert a deep recession.

Each time that the Fed raises interest rates, it increases the danger that the slump will deepen. The central bank has boosted interest rates, several times in the past few months, the last on July 20.

The Carter administration accepted yesterday's move with little reaction. A spokesman for Treasury Secretary G. William Miller said the move was "understandable" in light of current conditions.

The discount rate is the interest the Fed charges on loans to member banks, and is the most vsible signal of changes in money and credit policies. The board is independent of Congress and the administration.

The announcement came as the government published fresh economic statistics that showed the downturn so far is still a mild one -- a factor that may have contributed to the Fed's decision yesterday.

The figures showed that the nation's industrial output, a key indicator of economic activity, declined only lightly in July, while housing starts and personal income remained relatively robust (See story on Page F1).

Yesterday's action on the discount rate followed a meeting on Tuesday of the Fed's policymaking Open Market Committee, at which members apparently decided that more tightening was needed.

The Fed intervened in the New York money markets on Wednesday to push interest rates up further, and that same day the Chase Manhattan Bank led an industrywide move to boost the prime lending rate to a record 12 percent high.

The prime rate is the interest rate that private banks charge their most creditworthy corporate customers. The prime reached 12 percent only once before -- in 1974, near the start of one of the nation's worst recessions.

Although yesterday's Fed action was destined to bring the discount rate into line with other interest rates, rather than to lead any new push toward further tightening, it did confer the central bank's blessing on the increases so far.

The discount-rate rise yesterday reflected growing concern over continuing high inflation and the further weakening of the dollar -- the same two factors that sparked the previous rise in July.

The board also expressed dismay that the nation's money supply -- closely watched by Wall Street as an indicator of the anti-inflation fight -- might be increasing too rapidly. The Fed reported yesterday a sharp rise in the most recent week.

The half-percentage-point increase in the discount rate was large by historical standards, but generally in line with those the board has been authorizing in recent years, in the face of unusually rapid inflation.

The move was expected to be hailed in the foreign exchange markets, where the dollar has been slumping in recent days. Foreigners view increases in interest rates here as evidence of the Fed's determination to fight inflation.

The increase in the discount rate becomes effective today. The boost was requested by the Federal Reserve banks of New York, Philadelphia, Richmond, Cleveland, Kansas City and St. Louis.