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 $ζ(3)$ to produce non-trivial harmonic sum identities and the reduction of the resulting congruences between harmonic sums via a congruence between the Apéry numbers and another Apéry-like sequence.