Solder the unattached outdoor probe to the two solder pads that you removed the indoor thermistor from (red arrow).  For identification purposes, the blue arrow shows where the original outdoor probe wires are attached.  At this point, this wire has nothing on its other end.
 

Since there was only one wire leaving the case in its stock form, we will need to provide a slot (red arrow) for the new lead to exit the case.  I used a small, round (rat tail) file to form the slot.  The blue arrow points to the area of the liquid crystal display that contacts the fingers on the circuit board.  When you are reassembling the case, it is important that good contact is made.
 

Install the seven small screws, making sure that the two center screws below the display are snug (but not too tight) before tightening the rest of the circuit board screws.  Using an "X" pattern to tighten the screws will help insure that the circuit board is tightened evenly.  Slide the knot for the new wire you soldered in so that there is not too much loose wire inside the case and tighten the knot a bit.  There is no need to make this knot too tight.  It is just there to prevent the wire from tugging on your soldered connections.
 

This is the toughest part of the modification.  After stripping the ends of the probe wire and threading a couple pieces of heat shrink tubing on the thermistor leads, you get to solder the thermistor to the probe wire.  The reason that this is tough is that it either requires 3 hands or a way to keep the leads in contact with each other as you solder them.  Twisting the ends of the wires together before trying to solder them helps, but 3 hands works better.  Now slide the heat shrink over the soldered joints and shrink it with some heat from a lighter or the metal barrel of the soldering iron.
 
 

It's hard to see in this picture, but I added a couple more pieces of heat shrink to hold the wires together.  This is optional.  Notice that the thermistor is pretty small and the wires are very thin.  The wires happen to fit nicely between the socket 370 / socket 7 - socket and the processor, so you can monitor the processor's temperature from the back of the chip.
 
 

I've heard comments that some think that the thermometers are not very accurate.  This may be true if you compare this inexpensive unit to thermometers costing much more.  However,  a couple of degrees off throughout the range is more than accurate enough to tell what the processor or heat sink are doing at a particular time.  The displayed temperature of the two probes measuring the same temperature does vary.  However, as you can see, at a rather chilly room temperature, the difference between my $15 thermometer and a calibrated Omega thermocouple thermometer (thanks again Charlie!) using a type "K" thermocouple is negligible.  At 120°F the RS thermometer is off by a few more degrees.

So there you have it.  No more excuses that your socket 7 has no temperature monitoring.  If you want to know the temperature of the processor's case, you have the means.