  We investigate the spectral properties of Schr\"odinger operators in l^2(Z)
with limit-periodic potentials. The perspective we take was recently proposed
by Avila and is based on regarding such potentials as generated by continuous
sampling along the orbits of a minimal translation of a Cantor group. This
point of view allows one to separate the base dynamics and the sampling
function. We show that for any such base dynamics, the spectrum is a Cantor set
of positive Lebesgue measure and purely absolutely continuous for a dense set
of sampling functions, and it is a Cantor set of zero Lebesgue measure and
purely singular continuous for a dense G_\delta set of sampling functions.
