* Euro zone yields rise Draghi cites "vigorous" inflation
* Money markets increase bets on second 2019 rate hike
* Euro up, stocks fall on Draghi speech (Updates pricing)
LONDON, Sept 24 (Reuters) - Germany's 10-year Bund yield rose on Monday to its highest since mid-June after European Central Bank President Mario Draghi highlighted a "vigorous" pick-up in inflation and noted building wage pressures across the bloc.
The inflation recovery depends on interest rates staying low through next summer, Draghi said. But he called an acceleration in underlying euro zone inflation "relatively vigorous" and expressed confidence that a rise in wage growth would continue .
His statement pushed up bond yields across the euro zone and led money markets to increase their bets that interest rates would rise a second time by the end of 2019.
Draghi said that while annual headline inflation was likely to reach 1.7 percent each year until 2020, he saw "a slowing contribution from the non-core components of the general index, and a relatively vigorous pick-up in underlying inflation."
That would see inflation, excluding food and energy, reaching 1.8 percent in 2020, he added.
The comments sent Germany's 10-year yield as high as 0.51 percent, its highest point since June 12. The five basis-point move puts it on track for its biggest one-day rise since August 1. Ten-year French yields also hit a mid-June peak, rising nearly five basis points to 0.83 percent, its biggest one-day rise in almost two-months.. Yields also jumped in other "core" markets, such as Netherlands and Austria.
German two-year yields, which tend to be more sensitive to interest rate moves, also rose to the highest level since March, up two basis points to minus 0.53 percent.
"The 'vigorously' is strong language. Even if he said 'relatively vigorously', markets picked up the vigorous part way more than the relative," said Arne Petimezas, an analyst at AFS Group in Amsterdam.
Martin van Vliet, senior rates strategist at ING Bank, also said "vigorously" was the "key phrase."
"It is a notable move in markets and I am curious why he decided to strike this fairly hawkish tone," van Vliet said.
In southern Europe, Italian yields also extended their rise across the curve, with 10-year yields hitting 2.96 percent, their highest in over two weeks. . Five-year Italian bonds were set for their biggest one day rise since August 13, up 13 basis points on the day..
The spread between Italian and German yields hit a one-week high of 243.7 basis points.
The ECB has pledged to keep rates steady "through the summer of 2019," and markets have only recently started fully pricing a rate rise next October. Draghi's statement saw euro zone money markets pushing up bets on next year's rate hikes.
The difference between the overnight bank-to-bank interest rate for the euro zone (Eonia) and forward Eonia rates dated for the ECB's December 2019 meeting rose as much as 17 basis points compared with 15 bps earlier in the day.
That means markets are not only fully pricing in a 10-basis- point rate hike in October, but pricing also suggests investors are ratcheting up expectations for a second hike by end-2019.
A market gauge of longer-term inflation expectations for the euro zone rose to 1.7 percent after Draghi's speech. The five-year, five-year break-even forward rate which is tracked by the ECB is now at levels last seen three weeks ago.
The euro rose 0.6 pct at one point, surpassing $1.18, to its highest versus the dollar since June 14.
Euro zone stocks also slipped to trade 0.5 percent lower on the day. Interest rate-sensitive banks briefly turned higher but later reversed the advance to end down around 1 percent.
(Reporting by Virginia Furness, Dhara Ranasinghe, Danilo Masoni and Tom Finn Editing by Matthew Mpoke Bigg)