On Friday, it stood at $1.3640 at 1230 GMT after briefly rising above $1.3660, while trading volumes dried up on New Year's Eve.
Japanese markets were fully closed, and markets were expected to close early in parts of Europe and the United States.
"Looking at euro/dollar, we are witnessing the structural weakness of the dollar," said Paul Mackel, currency strategist at ABN Amro in London. "Next week the market is going to focus on how far will this go in 2005."
The yen rebounded broadly, moving away from record lows versus the euro set at 141.61 yen earlier this week, to trade at 139.70 yen.
Against the dollar, it gained about two thirds of a percent to 102.45 yen, climbing after a fall earlier this week in the wake of the deadly tsunami around south Asia.
Some 124,000 people were confirmed killed in the disaster.
"There was a knee-jerk reaction in the yen following the tragic event and we are seeing a pull back now," said Kamal Sharma, currency strategist at Dresdner Kleinwort Wasserstein in London.
Investors' major concern is whether the United States can continue to attract enough foreign capital to plug its current account deficit, which is now running at 5.6 percent of gross domestic product.