  We demonstrate how to approximate one-dimensional Schr\"odinger operators
with $\delta$-interaction by a Neumann Laplacian on a narrow waveguide-like
domain. Namely, we consider a domain consisting of a straight strip and a small
protuberance with "room-and-passage" geometry. We show that in the limit when
the perpendicular size of the strip tends to zero, and the room and the passage
are appropriately scaled, the Neumann Laplacian on this domain converges in (a
kind of) norm resolvent sense to the above singular Schr\"odinger operator.
Also we prove Hausdorff convergence of the spectra. In both cases estimates on
the rate of convergence are derived.
