Hmm... constant 1v?
Well, you can't use a zener because the lowest voltage used with Zeners is 1.8v.
You could put a normal rectifier diode in series with a Schottkey. Normal rectifier is 0.7v and Schottkey is 0.3v. The voltage will rise a bit as current rises though, it is not extremely tightly controlled. Temp changes it a small amount too.
But you mention LEDs? So I assume you're doing that common emitter circuit. For this, so you see the tricky problem in these. You need either a really specific driving voltage or a really specific low ohm resistor.