  We obtain an estimate for the cubic Weyl sum which improves the bound
obtained from Weyl differencing for short ranges of summation. In particular,
we show that for any $\varepsilon>0$ there exists some $\delta>0$ such that for
any coprime integers $a,q$ and real number $\gamma$ we have \begin{align*}
\sum_{1\le n \le N}e\left(\frac{an^3}{q}+\gamma n\right)\ll (qN)^{1/4}
q^{-\delta}, \end{align*} provided $q^{1/3+\varepsilon}\le N \le
q^{1/2-\varepsilon}$. Our argument builds on some ideas of Enflo.
