  We study perturbations of the self-adjoint periodic Sturm--Liouville operator
\[
  A_0 = \frac{1}{r_0}\left(-\frac{\mathrm d}{\mathrm dx} p_0 \frac{\mathrm
d}{\mathrm dx} + q_0\right) \] and conclude under $L^1$-assumptions on the
differences of the coefficients that the essential spectrum and absolutely
continuous spectrum remain the same. If a finite first moment condition holds
for the differences of the coefficients, then at most finitely many eigenvalues
appear in the spectral gaps. This observation extends a seminal result by
Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we
show that the band edges are no eigenvalues of the perturbed operator.
