  We show that the one-dimensional periodic Zakharov system is globally
well-posed in a class of low-regularity Fourier-Lebesgue spaces. The result is
obtained by combining the I-method with Bourgain's high-low decomposition
method. As a corollary, we obtain probabilistic global existence results in
$L^2$-based Sobolev spaces. We also obtain global well-posedness in
$H^{\frac12+} \times L^2$, which is sharp (up to endpoints) in the class of
$L^2$-based Sobolev spaces.
