This theorem was thought to be proven by Max Dehn (1910), but Hellmuth Kneser (1929, page 260) found a gap in the proof.
The status of Dehn's lemma remained in doubt until Christos Papakyriakopoulos (1957, 1957b) using work by Johansson (1938) proved it using his "tower construction".
Papakyriakopoulos proved Dehn's lemma using a tower of covering spaces.
Whitehead (1958) gave a substantially simpler proof, proving a more powerful result.
Their proof used Papakyriakopoulos' tower construction, but with double covers, as follows: