Stevecim is right but you need to use a boolean difference to cut the band. not the merge

I find that if you first join the cutting lines check they are on the same plane (ie thickness 0) then extrude them through the object. this often gives a better cut, also you can visually check the it goes right outside the object to be cut.

Your posted 3dm has the parts chopped into 12 or more parts (was this from the aborted cutting)
cheers
shayne