  We prove a supercongruence modulo $p^3$ between the $p$th Fourier coefficient
of a weight 6 modular form and a truncated ${}_6F_5$-hypergeometric series.
Novel ingredients in the proof are the comparison of two rational
approximations to $\zeta (3)$ to produce non-trivial harmonic sum identities
and the reduction of the resulting congruences between harmonic sums via a
congruence between the Ap\'ery numbers and another Ap\'ery-like sequence.
