The NYT should have pointed this fact out in a discussion of the crisis facing the eurozone. The article includes a quote from Wolfgang Schäuble, Germany's Finance Minister:
"I’m convinced that if we abandoned the promise of euro stability, we would have a few weeks, maybe a few months of relief on the financial markets. But after a few months the problem would return. It is all about trust.”
If the European Central Bank (ECB) is buying debt issued by eurozone countries and explcicitly guaranteeing their value, then it is logically impossible for the problem to return in a few months. The problem at the moment is one of solvency, Greece, Italy, and now Spain and possibly France risk being in a situation where they cannot pay their debts. They effectively face bankruptcy.
However, their debt is payable in euros. The ECB will never run out of euros, it can create an infinite amount of euros. This means that Mr. Schäuble is either confused or being deliberately misleading when he claims that the problem would return if the ECB took responsibility for backing up sovereign debt. It will not.
In principle, the eurozone countries would risk inflation by going this route, but given the vast amounts of excess capacity throughout the eurozone, this is not a near-term fear. (Of course, somewhat higher inflation would be desirable since it would reduce debt burdens and facilitate the adjustments of Spain, Greece, and Italy, so that they could regain competitiveness in the euro zone.)
The eurozone countries need not fear a flight from the euro for the same reason that the United States need not fear a flight from the dollar. If the euro were to plunge against the dollar and other currencies, the products of eurozone countries would become hyper-competitive. Their exports to the rest of the world would soar and imports would plunge. The United States, China, and other major countries would not tolerate this disruption to their economies. They would feel the need to support the euro and keep such a plunge from happening, in the unlikely event that the markets sent the euro tumbling.