WASHINGTON — Home sales in December reached their highest pace in nearly a year. The gain coincided with other signs that the troubled housing market improved at the end of last year.

Analysts cautioned that sales remain historically low and that it will take years for the home market to return to full health.

Still, the third straight monthly sales increase was encouraging. And economists noted that conditions are in place for further gains this year:

Prices have declined. Mortgage rates have never been lower. Homebuilders are slightly more hopeful because more people are saying they might be open to buying this year. And home construction picked up in the final quarter of last year.

“There’s no denying that home sales are still very low and will remain low for a few years,” said Paul Dales, an economist with Capital Economics. “But after having risen in each of the last three months ... it is clear that a housing recovery is now well under way.”

Sales of previously occupied homes rose 5 percent to a seasonally adjusted annual rate of 4.61 million in December, the National Association of Realtors said Friday. It’s the best level since January 2011.

For all of 2011, sales totaled only 4.26 million. That’s up slightly from 4.19 million in the previous year. But it’s far below the 6 million that economists equate with healthy housing markets. In 2005, at the peak of the boom, 7.1 million homes were sold.

Hiring has improved, which is critical to a housing rebound. Fewer people sought unemployment benefits last week than at any time in nearly four years.

The median sales price of a previously occupied home rose 2.3 percent from November to December to $164,500.

The supply of homes has declined, though it’s still historically high at 2.38 million. At last month’s sales pace, it would take nearly seven months to clear those homes.

If the supply continues to fall, prices could rise, more sellers would put homes on the market and more people would likely consider buying, said Pierre Ellis, an analyst at Decision Economics.