Many analysts worry the eurozone will enter a deflationary spiral
The eurozone's annual rate of inflation turned negative in June for the first time since the single currency was introduced in 1999.
Prices in the 16-nation zone fell 0.1% in the past year, Eurostat said. The inflation rate had been 0% in May.
Inflation in the eurozone has been dragged down by lower energy and food prices, and by falling demand for goods from companies and households.
The European Central Bank's target rate for inflation is just below 2%.
Some analysts fear that this is the start of a period of deflation for the eurozone.
Deflation is considered damaging to an economy as consumers tend to delay making purchases until prices fall further. Without consumer spending to stimulate growth, economic output falls.
"There are plenty of reasons to believe that the annual decline of 0.1% in June is just the beginning of a downward trend," said Daniele Antonucci from Capital Economics.
"At this stage, we expect negative inflation rates for the next six months or so. With factory gate prices falling, wage growth likely to slow sharply and a big amount of spare capacity in the economy, core inflation will decelerate considerably."
But European Central Bank officials have played down the threat of deflation, saying that any fall in prices is likely to be temporary and linked to the drop in oil price.
Inflation in the eurozone peaked at 4.1% in July last year when oil reached a record high of $147 a barrel.
The price of oil has since fallen back to about $70 a barrel.