Date: 5/8/2008 1:19 pm EDT
Neatly tile up to within 1/8" of the existing base board and caulk that small 1/8" gap. Don't use the grout there, use caulk instead.
In fact, you can buy colored caulks that will match standard portland cement grouts. Your grout is not a portland cement grout but you still may be able to match one of those tile caulks to your Avaire urethane grout.