  A difference basis with respect to $n$ is a subset $A \subseteq \mathbb{Z}$
such that $A - A \supseteq \{1, \ldots, n\}$. R\'{e}dei and R\'{e}nyi showed
that the minimum size of a difference basis with respect to $n$ is
$(c+o(1))\sqrt{n}$ for some positive constant $c$. The best previously known
lower bound on $c$ is $c \geqslant 1.5602\ldots$, which was obtained by Leech
using a version of an earlier argument due to R\'{e}dei and R\'{e}nyi. In this
note we use Fourier-analytic tools to show that the Leech--R\'{e}dei--R\'{e}nyi
lower bound is not sharp.
