  We consider a perturbed Hill's equation of the form $\ddot \phi +
  (p_{0}(t) + \epsilon p_{1}(t)) \phi = 0$, where $p_{0}$ is real analytic and
periodic, $p_{1}$ is real analytic and quasi-periodic and $\eps$ is a ``small''
real parameter. Assuming Diophantine conditions on the frequencies of the
decoupled system, i.e. the frequencies of the external potentials $p_{0}$ and
$p_{1}$ and the proper frequency of the unperturbed ($\epsilon=0$) Hill's
equation, but without making non-degeneracy assumptions on the perturbing
potential $p_{1}$, we prove that quasi-periodic solutions of the unperturbed
equation can be continued into quasi-periodic solutions if $\epsilon$ lies in a
Cantor set of relatively large measure in $[-\epsilon_0,\epsilon_0]$, where
$\epsilon_0$ is small enough. Our method is based on a resummation procedure of
a formal Lindstedt series obtained as a solution of a generalized Riccati
equation associated to Hill's problem.
