There are two solutions that I know of, but I think the basic idea could be extrapolated into many more.
Larry is correct in that the solution has something in common with a Mobius strip. But as mentioned in my response to Larry's proposed solution, I believe that (like a Mobius strip) the cut needs to start and end in the same place to result in a continuous loop.
And the 'meat' of the hint? The cut need not go all the way through.