  We construct examples of potentials $V(x)$ satisfying $|V(x)| \leq
\frac{h(x)}{1+x},$ where the function $h(x)$ is growing arbitrarily slowly,
such that the corresponding Schr\"odinger operator has imbedded singular
continuous spectrum. This solves one of the fifteen "twenty-first century"
problems for Schr\"odinger operators posed by Barry Simon. The construction
also provides the first example of a Schr\"odinger operator for which M\"oller
wave operators exist but are not asymptotically complete due to the presence of
singular continuous spectrum.
