Francis said he understood the motivation behind Marx's offer to resign but would not accept it.
“That is my answer, dear Brother. Continue as you suggest, but as Archbishop of Munich,” Francis told Marx in the letter.
A former head of the Catholic Church in Germany, Marx is not under any suspicion of having taken part in abuse or cover-ups.
The church is investigating abuse allegations in another German archdiocese, Cologne, after a report in March found hundreds of victims there.