Guardiola flew home to Spain on Sunday to be with his family for Christmas and has given no indication of his future plans, having been heavily linked to Manchester City.
He will return in January when the Bavarian giants fly to Doha for a training camp before the season resumes on January 22 when they play Hamburg away.
"We appreciate everything Guardiola gave to our club and hope to celebrate a lot of success this season, especially as we now know he is leaving," Rummenigge said in a club statement.
"He is a calm, balanced professional who can deal with stars and lets you play a flexible style of football -- we have found what we were looking for.
"With Ancelotti, we have a new, successful trainer for Bayern. We are looking forward to working together."
The 56-year-old Ancelotti will bring a wealth of experience to Bayern having won the Champions League twice with AC Milan in 2003 and 2007, then with Real Madrid in 2014.
"It's a great honour for me that I will be trainer of such a great club as FC Bayern next season," Ancelotti, who has signed a three-year contract until 2019, wrote on Twitter.
"I want to wish my friend Pep Guardiola and Bayern Munich all the best for the rest of the season."
Ancelotti will inherit a star-studded squad, although he admitted in October that he does not enjoy watching Bayern play given their domination of Germany's top flight.
"Bayern will win the Bundesliga (this season) without even taking their hands out of their pockets," said the Italian.
"I must confess that I cannot enjoy watching Bayern's games.
"There's just not enough competition for them."
Having joined Bayern in 2013 to much fanfare, Guardiola has so far won five trophies with the Bavarian giants, giving him a career total of 19 as coach following his 14 in four years from 2008-2012 at Barcelona.
Guardiola's side are still in the hunt for a repeat of their 2013 treble of Champions League, Bundesliga and German Cup, and will face last season's finalists Juventus in the last 16 of Europe's premier club competition.
Guardiola's tenure at Bayern is likely to be judged on his Champions League form as he bids to reach the final in Milan on May 28 having exited at the semi-final stage the last two seasons.
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