At what depth does the temperature remain relatively stable?

That would be the "frost line", which varies around the world.

Around where I live (Central Ohio), the rule of thumb minimum for burying plumbing lines is 16".

Rick,

Sorry I didn't express myself correctly. The idea is to run water pipes down into the ground where the temp is a constant 50°F or so? What I needed to know is at what depth should these pipes be placed?

Cheers

Colin

That point would be below the frost line... so in my area we put water lines at least 16" down.