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* EU leaders choose France's Lagarde as next ECB chief
* Euro zone bond yields hit record lows
* Stock markets open firmer
* Markets relieved that a hawk will not lead ECB
* Lagarde expected to continue Draghi's dovish stance (Updates with more details on price action, adds comment)
LONDON, July 3 (Reuters) - Euro zone bond yields hit new record lows on Wednesday after a decision by European Union leaders to pick France's Christine Lagarde as next European Central Bank chief reassured markets that the central bank's dovish stance would remain in place.
In a week where comments from even some of the ECB's hawkish members hinted at more easing if inflation remains unacceptably low, the decision on the next ECB chief only added fuel to a stellar rally in government bond markets.
Analysts expect Lagarde to continue current ECB chief Mario Draghi's dovish policy stance. If approved by the European parliament, Lagarde will succeed Draghi when his term expires at the end of October.
"We have to admit that we didn't see this coming, especially after she strongly denied being a candidate last year," said Frederik Ducrozet, a strategist at Pictet Wealth Management.
"She should provide continuity after Draghi, an important driving factor for markets. The ECBs reaction function is unlikely to change dramatically under Lagardes presidency."
Most 10-year bond yields across the bloc tumbled 3 basis points, with those in the Netherlands, Austria , Spain and Portugal all hitting record lows.
Belgium's 10-year bond yield fell into negative territory for the first time, joining an ever-growing pool of sub-zero yielding fixed income assets.
In Germany, the euro zone's benchmark government bond issuer, 10-year bond yields hit a record low at minus 0.397% -- now within a whisker of the same level as the ECB's deposit rate of minus 0.40%.
French 10-year bond yields pushed further into negative territory, hitting a new low of minus 0.09%.
The euro was steady in early European trade at $1.1283 , while stock markets in the bloc edged higher at the opening.
Analysts said the reaction in markets also reflected relief that EU leaders had not selected a more hawkish candidate such as Germany's Jens Weidmann to lead the ECB.
Andrew Kenningham, chief Europe economist at Capital Economics, which expects the ECB to cut rates in September and relaunch asset purchases before year-end, said the risk to that forecast was the possibility of a hawk succeeding Draghi.
"So it follows that the key thing about the appointment of Christine Lagarde as the next head of the ECB is that she is not Jens Weidmann," he said.
In Italy, 10-year bond yields slid 9 bps to a 14-month low of 1.76%.
Italy is seen as a key beneficiary of further ECB easing and its bond market has outperformed in recent days as comments from ECB officials fuel expectations of rate cuts and possibly a fresh round of asset purchases. (Reporting by Dhara Ranasinghe; Editing by Kevin Liffey)