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There will be relief more than celebration today that after 18 months the eurozone has eased out of recession.

The growth of 0.3% is modest, although slightly higher than expected, but it reflects growing confidence that the worst of the crisis lies in the past and that a recovery is taking hold.

Germany is leading the way. Its economy grew by a robust 0.7% in the three months up to June. All the signs are that not only investor confidence is returning but consumers are spending again, particularly on cars and household appliances. Germany very much remains the engine of the European economy.

France, more surprisingly, posted growth of 0.5%. That was twice as fast as expected and lifts France out of recession. A few days ago President Francois Hollande had said "it's still fragile and precarious but something is happening with the economy".

Today's figures reflect a change in sentiment in the past three months. Surveys of manufacturers have shown increased optimism. Industrial production in June grew at the fastest rate for three years. Some countries, like Spain, have seen a surge in exports. Although some of the economies in southern Europe are still shrinking, the rate of decline is slowing. Portugal grew by 1.1%.

But that is as good as it gets. There are grounds to be cautious.

Growth at 0.3% is tepid. Official European figures still see the eurozone economy contracting for the whole of 2013 before rebounding next year.

Although industrial production for June was up for the eurozone as a whole (up 0.7%) it was down in France, Spain and Portugal. In the Netherlands it was down 4.1%.

Slender growth will only have a marginal impact on unemployment. To make a significant difference to the jobless figures growth of 1.5% is needed. That is some way off. Some are saying that even by 2018 unemployment in Spain could still be 25%. France's jobless rate of 10.6% could still climb to 11.6% before coming down. High unemployment will continue to act as a brake on the wider economy.

Greece remains a problem unsolved. As was disclosed this week, the Bundesbank believes that Greece will need a new bailout next year. Although even here there are some green shoots, its economy shrank by 4.6% in the second quarter of the year. Since 2008 its economy has declined 23%, unprecedented in the modern era. Again the recession is predicted to moderate, but the country is currently not meeting its targets for raising tax revenue.

In Spain and Italy small and medium-sized businesses are still struggling with tight credit and both economies remain weak.

Many of these southern European countries are still having to reduce their deficits, even though austerity has morphed into austerity-lite, but spending cuts are still proving a drag on demand.

The eurozone crisis is ebbing slowly and some confidence is returning, but the recovery is fragile. What is unclear is whether the eurozone will be able to discover significant growth or whether the zone will flat-line and stagnate. More important is the question whether unemployment will start falling significantly or whether this will be a jobless recovery.

Confidence in the euro and the wider European project depends on being able to deliver jobs.