I like @Joe's solution actually, but I'll offer another.
You may be able to find another source besides the light.
In the attic, you may be able to find a live circuit you can tap into. Put a receptacle in the rafters (above the insulation), and then you can plug the transformer in there, and run a low-voltage wire down through the soffit to where the light is mounted. This is the most discreet option, as you will not see any wires or plugs at all, and your existing exterior light won't be affected.
You also could probably find somewhere inside the house to put the transformer, plugging it into an existing outlet, and then run the low-voltage wire up into the attic, and down to the soffit. Just keep in mind that the longer wire you have, the more of a voltage drop you'll get and the bigger a wire you will need.
You may also have a place where you can simply run a wire straight out through the wall (even if it's not directly where the camera is mounted - eg, if could even be at ground level), and then route the wire around the outside of the house (eg, hidden along the edge of the soffit, along window frames, edge of siding, etc) to the camera. I've done this before while adding satellite TV to houses, and done properly it can be completely hidden.
You may be able to do the same from the basement/crawlspace if you have one. Find a receptacle somewhere in the basement (or add one along an existing circuit), and then drill a 1/2" hole through the concrete, running a low-voltage line in conduit (PVC conduit is fine) out of that up to where the camera is mounted. Seal up the hole and again, you have a conduit running up, but it's low voltage and you have no transformer/receptacle visible. If you don't have a hammer drill now, this may be a much more difficult option, or a good excuse to buy a hammer drill, depending on how you look at it.
With all of these options, you get the benefit of no exposed outdoor transformer (and the transformer shown in the pic is definitely not designed to be exposed to the elements), and you can also plug the transformer into a UPS (battery backup) so that the camera will continue to work even if the power is out (which is only usful if the DVR or whatever you have it plugged into is also on a UPS, of course).