Protesters in Greece have objected to steep cuts in social services.
The OECD now forecasts that gross domestic product will decline by 0.1% next year in the eurozone, the 17-nation currency area. It had previously expected growth of 0.9%. The OECD also cut its 2012 forecast to -0.4% from -0.1%.
Unemployment in the region is forecast to rise to 11.9% in 2013 from 11.1% this year.
The previous OECD projection, issued in May, predicted unemployment would hit 10.8% in 2012 before spiking to 11.1% next year.
"The euro area, which is witnessing significant fragmentation pressures, could be in danger," wrote Pier Carlo Padoan, the organization's chief economist.
The danger of a messy end to the region's debt crisis is pushing up yields on sovereign debt, hurting Europe's banks and driving fears that a country could be forced to leave the continent's monetary union, Padoan said.
"Against such a fragile background, it is not difficult to imagine a situation in which something goes wrong," he said.
The diminished OECD projections for the eurozone are more optimistic than some private forecasts, but the organization is predicting a weaker performance than either the European Commission or International Monetary Fund, which see 2013 growth of 0.1% and 0.2% respectively.
Greece, the continent's most indebted economy, was granted a reprieve Tuesday as eurozone finance ministers came to terms with IMF negotiators over the next round of bailout money for the country. Still, Greece's debt will likely remain above 100% of GDP for at least the next decade.
Southern Europe's woes have already begun to affect the core. Germany has seen growth slow, and Moody's stripped France of its AAA credit rating earlier in November, citing its exposure to shocks from the eurozone periphery.
The eurozone problems, one of the biggest risks to the global economy, are spilling across borders and contributing to a global decline in consumer confidence, the OECD said. Emerging economies are particularly affected, and faring worse than their developed counterparts. Global trade, a key indicator of growth, is faltering.
The OECD sees slower growth for the United States as well, although the word's largest economy is facing fewer headwinds than Europe. GDP will increase this year by 2.2%, according to the OECD, which previously forecast growth of 2.4%. The group expects mediocre growth next year, and an expansion of 2.0%.
The biggest stumbling block for the U.S. is the so-called fiscal cliff, a set of automatic tax increases and spending cuts that will go into affect next year without congressional action. Many economists believe the U.S. will return to recession if policymakers fail to strike a deal.
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