  We consider Schr\"odinger operators on the real line with limit-periodic
potentials and show that, generically, the spectrum is a Cantor set of zero
Lebesgue measure and all spectral measures are purely singular continuous.
Moreover, we show that for a dense set of limit-periodic potentials, the
spectrum of the associated Schr\"odinger operator has Hausdorff dimension zero.
In both results one can introduce a coupling constant $\lambda \in (0,\infty)$,
and the respective statement then holds simultaneously for all values of the
coupling constant.
