The building society said that the average property was valued at £162,262 at the end of 2012, following a 0.1% drop in December.
Buyers nervous about the economy and a lack of mortgage availability meant there was relatively little housing market activity during 2012.
The 1% drop reversed a 1% average price increase in 2011, the Nationwide said.
However, the average price masked significant differences in property price changes in the different nations and regions of the UK.
For example, prices were 0.7% higher in London at the end of 2012 compared with a year earlier but dropped 8.2% over the same period in Northern Ireland.
The figures are based on the Nationwide's own mortgage data, and show less of a bubble in the capital than Land Registry figures released on Wednesday which indicated house prices in London had risen by 7% in the year to the end of October.
Outlook
The Nationwide said that conditions in the housing market remained "fragile", despite positive jobs news, with wages growing more slowly than the cost of living.
"The uncertain outlook for the wider economy is also likely to have kept many potential buyers on the sidelines, unwilling to make a major financial commitment until they feel more optimistic about the future," said Robert Gardner, the Nationwide's chief economist.
He said the reason that prices had remained "relatively resilient" in the circumstances, and had not dropped further, was that low interest rates had meant mortgages had remained affordable for many people.
This meant that people under financial pressure had not been forced to sell up, which would have increased supply and led to prices falling.
Mr Gardner predicted that activity would remain low in 2013, and prices would remain the same or dropping slightly throughout the year.
There have been slightly more optimistic forecasts from the Council of Mortgage Lenders (CML) and the Royal Institution of Chartered Surveyors (Rics), who have both predicted a modest rise in sales for 2013.
Completed sales were 6% higher during 2012 than they were the year before, according to figures from HM Revenue & Customs.
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