In reply to HammondR:
It'll probably depend on your general fitness and max grade. Afterall, Tom Randall led London wall (E5) in a pair of teenies approach shoes!
If you mean Grade III, it'll be heavier on your feet which can be great for ice!
If you mean grade V, and if that has been your max grade in normal boots then it is unlikely to be a great idea.
Usually people ask the contrary, can I ski in my climbing boots. It is always a compromise. You will always lose something out. If the main slant for the day is skiing, pick easyish route well within your grade and it shouldn't be a problem (BTW, you nice ski boots might get battered!).
If your main aim for the day is climbing, lug the extra pair (but find a way to keep them warm).
I have climbed grade IV ice in ski boots in the past... it is possible, I was glad it wasn't too long!