I would not want to pretend to know the best solution, but this sounds like you could use a power BJT in the linear region.
If you connect the gate to the power trough a resistor in series with your pot, then it should act nearly as a larger pot with a bit of a dead zone at the bottom. Well, and also only work for on current direction.
**broken link removed**
Please wait for someone more knowledgeable than me to confirm though, as I am still slightly unconfortable with BJTs.
As a side remark you will never get a perfect 0 with any transistor.