We prove that any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension, with periodic, even in space, initial data of small size \ep , is almost globally defined in time on Sobolev spaces, i.e. it exists on a time interval of length of magnitude \ep^{-N} for any N, as soon as the initial data are smooth enough, and the gravity-capillarity parameters are taken outside an exceptional subset of zero measure.