By Sumeet Desai and Fiona Shaikh

LONDON (Reuters) - House prices rose for the second month running in June, leaving them less than 10 percent down on a year ago, the Nationwide building society said on Tuesday, in another sign the market may be stabilising.

The mortgage lender said house prices rose 0.9 percent this month, taking the annual rate of decline to 9.3 percent -- the smallest fall since July 2008 -- from 11.3 percent in May.

And prices in the three months to June were 0.9 percent higher on the previous three months, the first positive reading since December 2007, with the average house price now standing at 156,442 pounds.

"House prices have risen in three of the last four months, suggesting that the improvement that began to show up in March represents more than just statistical noise," said Martin Gahbauer, Nationwide's chief economist.

Earlier, a survey by GfK/NOP showed consumer confidence hitting its highest level in 14 months in June as people became more optimistic about their finances -- the latest in a long line of indicators suggesting the worst is over for the economy.

But policymakers are still being circumspect and not declaring victory yet over Britain's first recession since the early 1990s -- figures due at 9:30 a.m. are expected to show the economy shrank 2.1 percent in the first three months of 2009.

Nationwide also remained cautious, pointing out the stabilisation in prices had come despite very low house purchase activity as there had been a corresponding drop in the stock of property for sale.

The latter is probably a result of potential sellers and builders holding on to their properties waiting for an upturn but Nationwide said that would have to change.