As a lot of things with this edition (and I personally love that), it depends.
It ends if you are incapacitated, if you don medium or heavy armor or
a shield, or if you use two hands to make an attack with a weapon. You
can also dismiss the Bladesong at any time you choose (no action
Case A: Going strictly by RAW
As is phrased in the rules, it doesn't state "specifically" that you cannot use one arm with one weapon each, so you could pull this off that way. "Rules" don't forbid it.
Case B: Attempting some interpretation
Obvioulsy, this is a subjetive interpretation, so you can dismiss it or think otherwise. But maybe it's worth some thinking. I also think this is one of the points of this edition "lack" of specification.
It seems that the whole idea of the BladeSong feature is that you need to have a certain freedom of movement, and also that you dont have both of your hands occupied at the same time (hence, the restriction of shield and using a weapon with two hands) so it depends on what do you think it's needed for this feature to work, or how it looks like when the character is using it, and THIS is what you should talk about with your DM.
Maybe you need to be gesturing something with the free hand all the time for it to work? Then you shouldn't be able to dual wield.
Maybe you need to be gesturing for MOST of the time, so you can maybe use some kind of projectile without breaking this gesture chain.
Maybe it's something more arcane, and for some unknown reason your hands cannot be "connected" by the same harmful object (hence, A weapon) or the "song" harmony will break. This way, you should be able to use dual wielding.
So, it's your call. Anyway, you should talk to your DM (and also to the rest of the party), so nobody feels like you are "cheating" and everybody can have a good time playing.