The euro fell further.
The main factors driving sentiment were an assurance from the ECB that it would counter deflationary pressures, and a hint from the US Federal Reserve that US rates may rise sooner than expected.
France, already on the back foot because of sluggish economic activity which has just forced the Socialist government to halve its growth outlook for the year to 0.5 percent, was hit by a new political crisis as markets opened.
President Francois Hollande told his Prime Minister Manuel Valls to form a new government, and Valls immediately offered the resignation of his administration.
The upheaval was triggered by a weekend speech by Economy Minister Arnaud Montebourg, attacking French, German and European Union austerity policies.
But despite this political shock, which would normally be expected to weigh down on French stocks and push up bond yields, the French markets were buoyant thanks largely to comments from the head of the European Central Bank, Mario Draghi.
In late morning trading, the French CAC 40 stock index was showing a gain of 0.93 percent to 4,292.42, although this was down from initial gains.
In Frankfurt, the DAX index of 30 leading shares was up 0.99 percent to 9,431.94 points
The interest or yield indicated by French 10-year bonds already issued and traded on the secondary market fell to 1.315 percent.
At brokers Credit Agricole CIB, economist Frederik Ducrozet said that the fall in the French bond yield could appear "counter intuitive", given the government crisis.
But the market was focused mainly on "comments by Mario Draghi," he said.
- ECB moves eurozone markets -
Draghi expressed confidence on Friday that action already taken, including record low interest rates, were enough to ward off any threat of deflation but that in any case the bank would ensure that deflation did not take hold, implying a possible injection of funds into the eurozone economy.
This pushed down eurozone bond yields, in some cases to record low levels.
The yield on the 10-year German bond, the benchmark for the eurozone, fell to a record low level of 0.944 percent, as did the Italian 10-year yield to 2.484 percent.
The yield in Spain fell to 2.290 percent and for Portugal to 3.087 percent.
The euro fell to $1.3199 from $1.3241 late on Friday, and to 137.29 yen from 137.60.
On Friday the head of the Federal Reserve Janet Yellen said that if the US employment market continued to strengthen at the current rate, the Fed might raise its interest rates sooner than investors had been expecting.
This attracted funds into the dollar, which rose to 104.02 yen from 103.87 late on Friday.
The euro was also undermined by a run of weak economic figures for the eurozone last week.
At Natixis bank in Paris, strategist Nordine Naam said that "the resignation of the French government is adding to a negative European climate, even though this is a marginal issue since it has not affected the euro."
Sterling rose against the euro to 79.61 pence and against the dollar to $1.6578.
The Swiss franc firmed to 1.2089 francs to the euro but slipped to 0.9159 to the dollar.