An S-unimodal map $f$ is said to satisfy the Collet-Eckmann condition if
the lower Lyapunov exponent at the critical value is positive.  If the
infimum of the Lyapunov exponent over all periodic points is positive
then $f$ is said to have a uniform hyperbolic structure.  We prove that
an S-unimodal map satisfies the Collet-Eckmann condition if and only if
it has a uniform hyperbolic structure.  The equivalence of several
non-uniform hyperbolicity conditions follows.  One consequence is that
some renormalization of an S-unimodal map has an absolutely continuous
invariant probability measure with exponential decay of correlations if
and only if the Collet-Eckmann condition is satisfied.  The proof uses
new universal bounds that hold for any S-unimodal map without periodic
attractors.