I have been trying for about a week now to get the wiring right on a new ceiling light fixture in my basement. I am replacing an old fixture, which only had one white and one black wire to connect. Extending from the ceiling box are two sets of white and black wires and a copper ground. On the original light fixture, white was connected to one of the white wires coming from the box and black to one house black. The remaining house wires -- one white and one black -- were twisted together and not capped.

The new light fixture has two sets of white and black wires, and a copper ground wire. I first tried to connect white to white (x2), black to black (x2), and the ground to the metal casing. The light turned on and off with the switch, but was very dim.

We called an electrician who came over and told us that one set of the black and white house wires was probably dead and to connect both white wires from the fixture to the "good" white house wire, and both black wires from the fixture to the "good" black house wire. We did this, capping the "bad" black and white house wires individually. The lights were on and bright when we turned the breaker back on. The problem is that now the light does not turn off with the switch. Should we twist the "bad" house wires together like they were with the old light fixture or what? I don't really like the idea of experimenting with wire connections, but I hate the idea of having to call another electrician back out to the house. Any help would be MUCH appreciated. Thanks!

First never, never, never ever, call that person again. I say person because he could not be an electrician.
I am a little supprised that he was smart enough to find his way to your house.

What you have is a fixture being fed with a switch loop. That is, power is coming to the fixture box on one of the cables (black and white pair) and then going to the switch and coming back on the other cable.

Connect the black wire (the good black), that you now have connected to the blacks of the fixture, to the white wire of the other cable. Put a wire nut on that connection.

While you are at it wrap the end of the white wire with black tape so the next time you will know that it is a "hot" wire.

Connect both fixture black wires to the remaining house black wire.
Connect both fixture white wires to the remaining house white wire.