Jane is pronounced Jeanne in French.

The "J" makes a "zh" sound, similar to the "s" in "measure" or "pleasure". 

The "a" is a short "ah" sound, similar to the "a" in "father". 

The vowel is nasalized, meaning air passes through the nose. 

Jane or Jeanne means “God is gracious” in Hebrew.

UPDATE-I asked my French teach how I should pronounce my name in French and she insisted I say:

J-A-N-E (jayn)

Voila!