  We estimate the spectral properties of the stochastic background of
gravitational radiation emitted by a cosmological population of hot, young,
rapidly rotating neutron stars. Their formation rate as a function of redshift
is deduced from an observation-based determination of the star formation
history in the Universe, and the gravitational energy is assumed to be radiated
during the spin-down phase associated to the newly discovered r-mode
instability. We calculate the overall signal produced by the ensemble of such
neutron stars, assuming various cosmological backgrounds. We find that the
spectral strain amplitude has a maximum $\approx (2-4)\times 10^{-26}
{Hz}^{-1/2}$, at frequencies $\approx (30-60)$ Hz, while the corresponding
closure density, $h^2 \Omega_{GW}$, has a maximum amplitude plateau of $\approx
(2.2-3.3) \times 10^{-8}$ in the frequency range $(500-1700)$ Hz. We compare
our results with a preliminary analysis done by Owen et al. (1998), and discuss
the detectability of this background.
