  Let $X, X_1, X_2,\ldots $ be a sequence of non-lattice i.i.d. random
variables with ${\bf E} X=0,$ ${\bf E} X=1,$ and let $S_n:= X_1+ \cdots+ X_n$,
$n\ge 1.$ We refine Stone's integro-local theorem by deriving the first term in
the asymptotic expansion for the probability ${\bf P} \bigl(S_n\in
[x,x+\Delta)\bigr)$ with $x\in\mathbb R,$ $\Delta >0,$ as $n\to\infty$ and
establishing uniform bounds for the remainder term, under the assumption that
the distribution of $X$ satisfies Cram\'er's strong non-lattice condition and
${\bf E} |X|^r<\infty$ for some $r\ge 3$.
