  We obtain a uniform ergodic theorem for the sequence $\frac1{s(n)}
\sum_{k=0}^n(\varDelta s)(n-k)\,T^k$, where $\varDelta$ is the inverse of the
endomorphism on the vector space of scalar sequences which maps each sequence
into the sequence of its partial sums, $T$ is a bounded linear operator on a
Banach space and $s$ is a divergent nondecreasing sequence of strictly positive
real numbers, such that $\lim_{n\rightarrow+\infty} s(n+1)/s(n)=1$ and
$\varDelta^qs\in\ell_1$ for some positive integer $q$. Indeed, we prove that if
$T^n/s(n)$ converges to zero in the uniform operator topology, then the
sequence of averages above converges in the same topology if and only if 1 is
either in the resolvent set of $T$, or a simple pole of the resolvent function
of $T$.
