We are adding a new fixture to an existing switch, which has only one lighting fixture currently in place. The home was built around 1965. How is the circuit supposed to be completed?

The easiest way (electrically) is to run the wires from the new fixture into the box for the old fixture.

One terminal of the old fixture should be connected to a set of white wires, and the other terminal should be connected to a black or red wire. (or maybe a white wire with a piece of black tape wrapped around it)

Connect the wires to the new fixture to the same places.