The French word in turn comes from Arabic yasamin, which is borrowed from Persian yasmin. The word is also found in Middle Persian: yasaman.
The Greeks also borrowed this Persian word: iasme, which the medical writer Dioscorides used in reference to a Persian perfume, perhaps oil of jasmine.
From Abraham Cowley's Essays in Verse and Prose (1669), there are these lovely lines:
Who, that has Reason, and his Smell,
Would not among Roses and Jasmin dwell?
Rather then all his Spirits choak
With Exhalations of Durt and Smoak?
The image is from Wikipedia: