LONDON — A rare full-length portrait by Gustav Klimt sold for 24.8 million pounds with fees, or about $39 million, at a Sotheby’s auction in London on Wednesday.

Sotheby’s had placed an estimate of £12 million to £18 million (about $19 million to $28 million) on Klimt’s 1902 portrait of Gertrud Loew, the 19-year-old daughter of Dr. Anton Loew, the director of a prominent Viennese private sanatorium that treated fin-de-siècle luminaries such as Mahler, Ludwig Wittgenstein, and Klimt himself.

The portrait, which had never previously been offered on the open market, was won by an unknown buyer...