Curve modifier wierdness

I’m having an issue with the curve modifier, trying to deform a cone along a nurbs curve. Below are before (left) and after (right) views of applying the curve to the dragon’s horn:

http://www.fylo.net/stuff/curve_wierd.jpg

The horn is positioned (loc and rot) where I want it. The horn and the curve have their object centers at the same location, as is the first point on the curve.

What am I missing?