Just soldering the wire won't work. I've salvaged a few of these RX's with broken antenna wires in three ways, but be warned you'll probably loose a good bit of range.
For either method you'll have to open the case and warm up your soldering iron then unsolder the stub of wire left attached to the board. I highly recommend MG desoldering wick - makes it super easy. The desoldering wick at radio shack is a joke, the plunger style solder sucker isn't horrible...but MG wick is hands down the best way to remove solder out of all methods I've tried. (Well, a professional desoldering iron with vacuum pump is great...but out of my price range!)
Method 1) Just take a piece of bare wire 31mm long and solder it where the center conductor of the original coax was. This works and I've had no range issues with park fliers in a small field with this fix.
Method 2) If the original coax on the antenna is still in good shape you can strip back the insulation carefully, twist the shield off to one side, strip off a bit of the inner insulation and then solder the braid and conductor back like the original stub was. This is really tricky to do right though. It's very easy to miss a loose strand of wire from the braid and end up with a short. You also want as little of the conductor exposed as possible since it will try to act like an antenna and detune things. The first one I tried doing this to seemed to work...but range was VERY limited - like 100 yards. The second one I did this to I was more careful and seem to have almost full range still.
Method 3) Take an antenna from a different broken 2.4Ghz RX and swap it over. I did this with the antenna from a
3ch RX that failed after a couple of crashes (these will bind with a 9x TX and work well on park fliers...but there's a component on them that's only supported by two tiny wires and after a couple of hard crashes those wires can break so they aren't great air RX's) The one I did this to works and has better range than the one with just a piece of wire and the first one I tried repairing the antenna on.
You could also try taking something like this
http://www.hobbyking.com/hobbyking/store/uh_viewitem.asp?idproduct=16666&aff=687394 and cutting off the connector then soldering it on.
Or you could try making a dipole, or turnstyle, or other antenna - a google search will give quite a bit of info on antenna design and it's not all that difficult.