  In 1991, Ursell gave a strong form of Watson's lemma for the Laplace integral
\[\int_0^\infty e^{-xt}f(t)\,dt\qquad (x\rightarrow+\infty) \] in which the
amplitude function $f(t)$ is regular at the origin and possesses a Maclaurin
expansion valid in $0\leq t\leq R$. He showed that if the asymptotic series for
the integral as $x\rightarrow+\infty$ is truncated after $rx$ terms, where
$0<r<R$, then the resulting remainder is exponentially small of order
$O(e^{-rx})$. In this note we extend this result to include situations when
$f(t)$ has a branch point at $t=0$ and when $x$ is a complex variable
satisfying $|\arg\,x|<\pi/2$.
