  Observations of metal lines in $\lyal$ absorption systems of small H~I column
density and their ubiquitous nature suggest that the intergalactic medium (IGM)
was enriched to about $Z \sim 0.01 \> Z_{\odot}$ by a redshift $z \sim 3$. We
investigate the role of winds from small star-forming galaxies at high $z$ in
enriching the IGM. The existence of large numbers of small galaxies at high $z$
follows naturally from hierarchical clustering theories (e.g. CDM). For
analytical simplicity we assume that the galactic winds escape the galaxies at
a single characteristic redshift $z_{in}$, and we model the galactic winds as
spherical shock waves propagating through the IGM. We then calculate the
probability distribution of the metallicity of the IGM, as a function of time
(for different values of $z_{in}$), adopting plausible galaxy mass functions
(from Press-Schechter formalism), cooling physics, star-formation efficiencies,
gas ejection dynamics, and nucleosynthesis yields. We compare this expected
distribution with the observed distribution of metallicities in the Ly$\alpha$
forest at $z=3$, the metal poor stars in the halo of our Galaxy, and with other
observational constraints on such a scenario. We find that galactic winds at
high $z$ could have enriched the IGM to a mean metallicity of $Z \sim 0.01
Z_{\odot}$ at $z \sim 3$, with a standard deviation of the same order, if
$z_{in} \la 5$, and that this satisfies all the observational constraints.
