The board of the foundation of the Museion in the city of Bolzano voted to keep the work by the late German artist Martin Kippenberger, the museum said in a statement.
Earlier in August the pope had written a letter to Franz Pahl, the president of the Trentino-Alto Adige region that includes Bolzano, denouncing the sculpture.
It "has offended the religious feelings of many people who consider the cross a symbol of God's love and of our redemption," Pahl quoted the pope as writing in the letter.
Pahl himself has long opposed the display of "Zuerst die Fuesse" ("First the Feet" in German), even staging a hunger strike this summer and saying he would not seek re-election unless it was removed.
In a telephone interview with The Associated Press on Thursday, Pahl said he was outraged by the museum's decision to keep the work, which he claims "pokes fun at the Catholic population and offends religion and the pope."
The 1990 wooden sculpture shows the crucified frog nailed through the feet and hands like Jesus Christ. The frog, eyes popping and tongue sticking out, wears a loincloth and holds a mug of beer and an egg in its hands.
The museum said the 3-foot (1-meter) -tall sculpture has nothing to do with religion, but is an ironic self-portrait of the artist and an expression of his angst.
"With humor and a tragicomic sense, which belongs to art since the times of Greek tragedy, Kippenberger ... faces his condition of suffering, which he expresses in many works, also, for example, in a video in which he crucifies himself," the museum said in a statement.
Born in Dortmund, Kippenberger moved from painting and sculpture to work in all mediums, often combining elements of Neo-expressionism, Pop and Dadaism. His art has been displayed across the world, including Zurich, Paris, Jerusalem, London and New York.
He died in 1997, aged 43.
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