an addendum: Several of you especially praise the tune written by Norbert Schultze.
Norbert Schultze was born Jan 26th, 1911 and has at least seen alive his 90th birthday early this year. He has written many songs for the Nazi propaganda films. He has composed the tunes for 'beauties' as 'Bombs on England' (songtitles in my translation) and 'Forward towards the East' with the line 'Führer, give orders, we follow'. For his contributions to the Nazi regime he was interned by the Allied Forces at the end of the war and let free in 1946 as 'mere supporter' (Mitläufer).
After the war he wrote a couple of Musicals and tunes for songs. None of them became anything remotely like famous. His tune to Lili Marleen was written in 1938 (that is the date most sources give), for Goebbels wanted a marching tune for the song Lili Marleen which he considered too 'pacifist' for his taste. If you want to hear the original 'too pacifist' tune which Leip had composed click here and go to the Midi in the lower right corner (original score in the lower left).
It was a complete surprise to me that there was another, original tune to this song and that the world has to thank Goebbels of all men for ordering a new tune to this song better fitting the Nazis' idea of soldier's music. I hate the thought of having to be thankful to Goebbels for anything and only get comfort by the thought that his instincts about the song being 'too pacifist' proved to be correct and his decision to let it get recorded against his instincts was wrong (in his sense), that is right (for the world).