Is the 2V input DC? You'll need a DC to DC boost converter, your power requirements aren't that steep there are plenty of IC's out there that can do it. Linear makes some IC's that should fit your needs. If your input is AC (though I don't see how it can be) you can use a step up transformer and then a bridge rectifier, but at those voltages it's going tobe very inefficient. A good DC-DC boost converter will be 80% or better efficient.
I would hazard a guess that there are no IC boost regs that can hold 80% efficiency boosting 2v (0.6A) up to 3.7V @ 350 mA. The peak currents on the 2V supply would go through the switches. Like I said, it's theoretically possible but I think the efficiency would be pretty bad. 60% would actually be pretty good for this. I believe Maxim has the biggest selection of low voltage boost ICs.
So a bridge rectifier will get you 2.8 Volts DC and you will still need a low voltage DC-DC converter with all the same problems mentioned by bountyhunter. Where is the cockamamie requiremnt coming from?
So how about a 1:1.5 stepup transformer that gets you to 3VAC, a bridge rectifier gets you 4.24VDC and a 3-terminal LDO that brings it back to 3.7VDC -- simple and straightforward - eh?