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

A modular supercongruence for $_6F_5$: an Ap\'ery-like story