Unfortunately, though normally a 420W would be plenty, and normally one rail vs. two rails does not matter at all, I can tell just by the amperage ratings that you'll run into trouble.
1.) Alienware PSUs are rebranded junk PSUs. If you're really lucky it's a Delta. If you're just sort of lucky, it's an FSP. Likely it's a HIPRO or HEC or Bestec or some other crap.
2.) When you have a PSU with two unequal +12V rails that's a good indication that it was one of the really early ATX 2.0 designs, the ones that gave multiple +12V rails a bad name. One of those rails is probably dedicated solely to the CPU. But the other rail is dedicated to your graphics card, motherboard, hard drives, fans, optical drives, PCIe devices, everything that uses +12V power but the CPU. And often, in situations like that, it was the smaller rail that was running everything.
Now first off, a "rail" in a power supply is probably not what you think it is. A "rail" has no physical manifestation. The power all comes from the same +12V transistor. A rail is a group of wires coming off the PSU's PCB that is monitored by an OCP circuit. In English: it's just a protections thing, that shuts the power supply off if it detects too much current on one rail, to stop the power supply from burning if you put too much load on it.
In most modern power supplies, that doesn't matter at all because the rails are divided logically. But with an old unit like yours, you have 80% of the load on one monitoring circuit... And thus it's quite easy to go over the OCP's trip point and cause the PSU to shut off.
If he has *just* the 5770 and a couple of hard drives and doesn't overclock the graphics card, he should be OK, assuming that the PSU isn't some garbage HEC that's derated so it's a 200W power supply. You can even overclock the CPU as much as you want, since no CPU will use more than ~14A. But be careful adding additional hard drives or overclocking the graphics card, because you could trip the OCP under load.
Frankly, I recommend a new power supply. The Corsair CX400W is $40 after rebate at Newegg.com, and it would be a good choice.