  We prove that for every integer k>1 there is a simply connected rational
homology 5-sphere $\scriptstyle{M^5_k}$ with spin such that
$\scriptstyle{H_2(M^5_k,\bbz)}$ has order $\scriptstyle{k^2},$ and
$\scriptstyle{M^5_k}$ admits a Riemannian metric of positive Ricci curvature.
Moreover, if the prime number decomposition of $\scriptstyle{k}$ has the form
$\scriptstyle{k=p_1... p_r}$ for distinct primes $\scriptstyle{p_i}$ then
$\scriptstyle{M^5_k}$ is uniquely determined.
