  The Laplace-Beltrami operator in the curved M\"obius strip is investigated in
the limit when the width of the strip tends to zero. By establishing a
norm-resolvent convergence, it is shown that spectral properties of the
operator are approximated well by an unconventional flat model whose spectrum
can be computed explicitly in terms of Mathieu functions. Contrary to the
traditional flat M\"obius strip, our effective model contains a geometric
potential. A comparison of the three models is made and analytical results are
accompanied by numerical computations.
