The government's index of leading indicators--a guide to the future course of the economy--rose by 1 percent in June, the Commerce Department reported yesterday.

The June increase was the 11th in the past year but was the weakest since last November, the government report said. The index had climbed by 1.2 percent in May and 1.3 percent in April.

Presidential economist Martin S. Feldstein said the leading indicators report continues "to point to further recovery in the months ahead. . . . I think there's no indication of weakness in the economy at this point."

Other government figures have shown that the economy grew very strongly in the March to June period this year, with an overall growth rate of 8.7 percent at an annual rate recorded for the gross national product. The leading indicators are supposed to provide a gauge for the likely course of the economy in the future.

A separate government report yesterday said that productivity in the nation's businesses jumped by 4.3 percent during the second quarter of this year. Productivity--or output per hour--typically increases at the start of a recovery as businesses step up production and overtime. At the same time, the Labor Department report said, unit labor costs rose by a mere 0.2 percent during the second quarter as hourly compensation was up by just 4.6 percent, the smallest increase since late 1977.

Feldstein called the productivity report "an economist's dream." He added that the unit labor cost numbers were "a very encouraging sign . . . these figures are really an indication of a fundamental transformation of the economy."

An improvement in the claims for unemployment benefit contributed to most of the June increase in the leading indicators overall increase, the Commerce report said. Two important components--new orders for business equipment and consumer goods--deteriorated during the month.

The leading indicators are generally considered best as a guide to the direction of the economy and not to its strength. However, government spokesman Larry Moran said that "it would have been nice to have contracts and orders for plant and equipment and consumer materials moving up."

Analysts agree that the economy is likely to continue to recover strongly during the current quarter, although many expect a slower rate of growth after that. "Not much significance can be attributed to the slowdown in the rate of increase" in the indicators, according to Gordon Richards, director of economic analysis at the National Association of Manufacturers. He said that "the decline in new orders for business equipment is to be expected in view of high interest rates, underutilization of capacity and depressed profitability, factors which work against a recovery in capital investment."

The biggest cloud hanging over the recovery is the prospect of a further rise in interest rates that could choke off housing and other interest-sensitive sectors, economists say. Rates have already risen noticeably in recent weeks as the Federal Reserve has moved to tighten up on credit.

A rise in interest rates would likely dampen the hoped-for recovery in business investment and would keep the already strong dollar high--worsening the prospects for exporters.

Of the 10 leading indicators that were available for the June index, six contributed to the increase, the Commerce report said. In addition to the unemployment claims, these were the average workweek; net business formation; building permits; stock prices; and money supply in 1972 dollars.

An accompanying index for coincident indicators, which are supposed to measure what is now happening to the economy, rose by 0.8 percent in June, the fourth consecutive monthly increase. The May figure was up by 1.7 percent.