Step 1: Cat-5 used for the tether
I used six wires (three pair) for the sensor (Vs+, Vout, Gnd). I twisted colored pairs together to give me the three conductors.
Step 2: Solder the LM35
I then shrunk the wrap over the soldered joints using a torch (on it's lowest possible setting.)
Step 3: Copper Cladding
I was able to just squeeze the LM35 into the copper tubing, then I squished the tubing just above the sensor in the vise...being very careful not to squish the sensor.
Step 4: Operational Sensor...but not waterproof
That's 16.7 mV (16.7 C or 62 F) on the voltmeter. Everything seems to be working.