The unemployment rate for April was revised from 12.2 percent to 12 percent, which is allowed May's rate to post the record high.
The unemployment rate in the overall European Union, which has 27 countries, was unchanged, holding at 10.9 percent for the third consecutive month, the data agency said.
The data office estimated that 19.222 million people in the eurozone and 26.405 million in the European Union were unemployed in May with the number of jobless increasing 67,000 in the eurozone and 16,000 in the European Union.
Compared to a year earlier, however, there are 1.344 million more unemployed workers in the eurozone and 1.324 million in the EU, the data office said.
The lowest unemployment rate in Europe in March was posted by Austria, where the rate held steady at 4.7 percent.
In the largest economy on the continent, Germany, the unemployment rate dropped from 5.4 percent to 5.3 percent in the month.
The highest rate for May was posted by Spain at 26.9 percent. The last figure posted by Greece refers to March at 26.8 percent.