WASHINGTON (Reuters) - Groundbreaking on new U.S. homes rose in June to its fastest pace in over three years, lending a helping hand to an economy that has shown worrisome signs of cooling.

The Commerce Department said on Wednesday that housing starts rose 6.9 percent last month to a seasonally adjusted annual rate of 760,000 units. That was the highest rate since October 2008.

The reading, which is prone to significant revisions, was above the median forecast in a Reuters poll of a 745,000-unit rate.

The U.S. housing market, which collapsed six years ago, has been a relative bright spot in the economy this year, although it remains hobbled by a glut of unsold homes.

Federal Reserve Chairman Ben Bernanke said on Tuesday the sector has shown modest improvement.

On Wednesday, the Mortgage Bankers Association said applications for loans to buy homes jumped last week as the interest rate on 30-year mortgages fell to a record low. In recent weeks, other private groups have said signed contracts for home purchases rose sharply in May and home prices rose that month.

The Commerce Department's revisions to data on housing starts from prior months were also upbeat. Groundbreaking during May was revised up to a 711,000-unit pace from a previously reported 708,000 unit rate. April's reading was also revised slightly higher.

Still, the broader U.S. economy has looked much more wobbly of late, and the housing sector is now so small it provides limited support. Indeed, if the recovery fails and the country tips back into recession, housing would also suffer.

In Bernanke's testimony before lawmakers on Tuesday, he said the U.S. economic recovery was being held back by anxiety over Europe's debt crisis and the path of U.S. fiscal policy. Planned belt tightening by the U.S. government in 2013 would send the country into recession, he said.

While Bernanke offered few new clues about whether the U.S. central bank was moving closer to a fresh round of monetary stimulus, he said policymakers meeting later this month would be looking closely for signs of any stall in the recovery of the labor market.

The pace of hiring in the United States slowed sharply in the second quarter, as did growth in factory output. Retail sales have also flagged in recent months.

The Commerce Department's report showed the increase in housing starts in June was spread broadly throughout the sector, with groundbreaking for single-family homes up 4.7 percent. This segment accounts for most of the market. Starts for multi-family homes - one of the report's more volatile readings - rose 12.8 percent.

New permits for building homes dropped 3.7 percent in June to a 755,000 unit pace.

(Reporting by Jason Lange; Editing by Neil Stempleman)