Thanks for the info.
I can now recreate what you have done. I just draw a curve, and close it, no welding, the arc stays an arc. I then use extrude to create the surface and afterwards the texture works fine.
What I was actually doing to create the extruded shape was rather different, and it might have something to do with it.
I create an arc just as a guideline for a follow me. Then I create an Upright Rectangle, and follow it along the arc. The reason for this was of how my Ruby script work. I have to create a lot of components that follows basically the same curve, ie curved panels, with mouldings, etc. It seems that in that case, the resulting follow me creates the "segments" somewhat differently, and hence the texture problem. This might mean some rethink on my approach.