The lady from the title refers to Hedy Lamarr. However, although her character is referred to in the title of the film, she really is not the star of this picture. Instead, John Hodiak is the star and he does a very nice job playing dual roles--an American immigration agent as well as a stateless Hungarian.
When the film begins, a man is accidentally run over when escaping from someone. Why folks are chasing him isn't certain...and you learn more about this as the film progresses. It seems that this unknown dead man is an illegal alien...and clues on his body point to his having just been in Cuba. So, Peter (Hodiak) goes down to this island nation to investigate...with the help of the local police. The trail then leads to a smooth but deadly jerk, played very well by George Macready (he played snake-like characters very well). So how is pretty Hedy involved? Watch the film.
I think for Ms. Lamarr, this film was a letdown and didn't do a lot for her career. Now it isn't because it's a bad film at all...it's very good. But she isn't given a lot to do but look pretty...even, inexplicably, during a trek through the Everglades near the end of the film...and her makeup and hair look superb!!! I live in Florida and even if this had been in the middle of winter, she sure COULDN'T have looked that perfect! Well worth seeing...and an interesting and unusual plot.