The Nationwide says the number of house sales remains "very low"
UK house prices rose by 0.9% in June, according to the latest survey from the Nationwide building society.
It said this was the third rise in the past four months, and shrank the annual rate of decline to just 9.3%, from 11.3% in May.
The increase in prices during the past month means the average home now costs £156,442, which is £15,973 less than a year ago.
The Nationwide said the stabilisation of prices was a "welcome surprise".
Since their recent low point in February, of £147,746, average UK house prices as measured by the Nationwide have now risen by £8,696.
"House prices have now 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 the Nationwide's economist Martin Gahbauer.
"What is unusual about the recent trend reversal, however, is that it has taken place against a background of transactions activity that is still very low by historical standards," he added.
The Nationwide said the best measure of short-term trends was to compare the average price for the past three months with that for the previous three.
On that basis, prices were now 0.9% higher, the first time they have been on an upward trend since December 2007.
The building society said that if this pattern continued then this year would end with prices down by only "small single digits".
"This would represent a stark shift from trends seen at the turn of the year, when most indicators were pointing to a repeat of the large declines seen in 2008," it said.
Although there has been no let up in rationing of loans by mortgage lenders, the building society said potential sellers, and builders, were putting very few properties up for sale, which was bringing some equilibrium to prices.
But it warned against interpreting its latest data as the beginning of a sustained recovery in prices.
Abnormally low supply levels would not last for ever, it warned.
"[The] increase in the enquiry pipeline has not yet led to large increases in transaction volumes, because credit criteria remain significantly more restrictive than in the years leading up to the downturn," said Mr Gahbauer.
"Rising unemployment and associated job insecurity are also limiting the extent to which enquiries can translate into actual transactions," he added.
Last week, HM Revenue & Customs (HMRC) reported that completed house sales in the UK had risen again in May, to their highest level since last October.
And on Monday, the Bank of England reported that the number of mortgages approved by lenders, but not yet lent, had risen for the fourth month in a row in May.
This suggests that the revival in buying and selling seen this spring may continue into the summer.
"Nationwide house price data provides further evidence that the residential property market appears to have found a floor, at least for the time being," said Simon Rubinsohn, chief economist at the Royal Institution of Chartered Surveyors.
Not all measures of house prices are pointing in the same direction.
Although the rival survey from the Halifax has also detected a recent upturn in prices, more comprehensive figures from the Land Registry for England and Wales reported that prices had still fallen in May, by 0.2%.
This meant the annual rate at which prices had dropped was 15.9%.