If it were mine, i'd feel more comfortable with at least a 500-watt power supply, more headroom for the future. But since you mentioned the PSU is old, i guess that would do for the meantime, unless it starts giving you problems soon.
Although you can run that systen with a 370watt psu,I recommend a 550watt psu or greater.The system will strain that psu quite a bit and you'll likely have to replace it in a few months at most.Not to mention that when a psu goes,it can take other components with it,including the motherboard.In the end this costs you more money.I always go with a higher watt and higher amp psu to increase the psu life,and not risk killing other hardware.Goodluck.
The 27 amp 12 volt rail will be more than adequate for your system.
I estimate power usage:
CPU - about 8 amps @12 volts
GPU - 5 amps @ 12 volts
MB - no more than 2 amps (outside of the CPU, motherboards don't use much 12 volt power
1 HD & 1 optical - no more than 3 amps @ 12 volts
fans - unless you have a lot, 1 amp @ 12 volts.
That's less than 20 amps @ 12 volts = 240 watts max
RAM - about 6 amps @ 3.3 volts
Motherboard logic - 4 amps @ 5 volts
2 drives logic - no more than 3 amps @ 3.3 volts
Subtotal here - no more than 55 watts.
Total power consumption, less than 300 watts.
I over estimated in a few places. Murphy needs his share.
As people have already said, you should be fine. That is a high quality psu. If you were trying to use a crappy brand psu, then the answer would be no. I use a 380W Antec Earthwatts in a system with a X2 4200+ and a 9600GSO 384mb both overclocked and it runs great.
That PSU will be fine. However, if you plan to upgrade your video in the future I would seriously invest in maybe a nice 550W PSU just to ensure you have the juice to run a nice card if you choose. It's better to spend $30-$50 more now than spend $80-$100 for another PSU and have this one laying around. Just a thought.
You'll be fine so long as you don't run prime 95 and furmark at the same time to see how much juice the whole things sucks up . I would still recommend you get a more powerful unit later, as you will be stressing your current one, especially if you want to put a more powerful GPU.