  An asymptotic expansion for the generalised quadratic Gauss sum
$$S_N(x,\theta)=\sum_{j=1}^{N} \exp (\pi ixj^2+2\pi ij\theta),$$ where $x$,
$\theta$ are real and $N$ is a positive integer, is obtained as $x\rightarrow
0$ and $N\rightarrow\infty$ such that $Nx$ is finite. The form of this
expansion holds for all values of $Nx+\theta$ and, in particular, in the
neighbourhood of integer values of $Nx+\theta$. A simple bound for the
remainder in the expansion is derived. Numerical results are presented to
demonstrate the accuracy of the expansion and the sharpness of the bound.
