  The purpose of this note is to introduce a new method for proving the
existence of Sasakian-Einstein metrics on certain simply connected odd
dimensional manifolds. We then apply this method to prove the existence of new
Sasakian-Einstein metrics on $\scriptstyle{S^2\times S^3}$ and on
$\scriptstyle{(S^2\times S^3)# (S^2\times S^3).}$ These give the first known
examples of non-regular Sasakian-Einstein 5-manifolds. Our method involves
describing the Sasakian-Einstein structures as links of certain isolated
hypersurface singularities, and makes use of the recent work of Demailly and
Koll\'ar, AG/9910118, who obtained new examples of K\"ahler-Einstein del Pezzo
surfaces with quotient singularities.
