A scheme for approximating the kernel $w$ of the fractional $\a$-integral by a linear combination of exponentials is proposed and studied.

The scheme is based on the application of a composite Gauss-Jacobi quadrature rule to an integral representation of $w$.

This results in an approximation of $w$ in an interval $[\del,T]$, with $0<\del$, which converges rapidly in the number $J$ of quadrature nodes associated with each interval of the composite rule.

Using error analysis for Gauss-Jacobi quadratures for analytic functions, an estimate of the relative pointwise error is obtained.

The estimate shows that the number of terms required for the approximation to satisfy a prescribed error tolerance is bounded for all $\a\in(0,1)$, and that $J$ is bounded for $\a\in(0,1)$, $T>0$, and $\del\in(0,T)$.