  An $\mathcal{O}(N(\log N)^2/\log\!\log N)$ algorithm for computing the
discrete Legendre transform and its inverse is described. The algorithm
combines a recently developed fast transform for converting between Legendre
and Chebyshev coefficients with a Taylor series expansion for Chebyshev
polynomials about equally-spaced points in the frequency domain. Both
components are based on the FFT, and as an intermediate step we obtain an
$\mathcal{O}(N\log N)$ algorithm for evaluating a degree $N-1$ Chebyshev
expansion at an $N$-point Legendre grid. Numerical results are given to
demonstrate performance and accuracy.
