The rate of fast radio bursts (FRBs) in the direction of nearby galaxy clusters is expected to be higher than the mean cosmological rate if intrinsically faint FRBs are numerous. In this paper, we describe a targeted search for faint FRBs near the core of the Virgo Cluster using the Australian Square Kilometre Array Pathfinder telescope. During 300 h of observations, we discovered one burst, FRB 180417, with dispersion measure (DM) = 474.8 cm−3 pc. The FRB was promptly followed up by several radio telescopes for 27 h, but no repeat bursts were detected. An optical follow-up of FRB 180417 using the PROMPT5 telescope revealed no new sources down to an R-band magnitude of 20.1. We argue that FRB 180417 is likely behind the Virgo Cluster as the Galactic and intracluster DM contribution are small compared to the DM of the FRB, and there are no galaxies in the line of sight. The non-detection of FRBs from Virgo constrains the faint-end slope, α < 1.52 (at 68 per cent confidence limit), and the minimum luminosity, Lmin ≳ 2 × 1040 erg s−1 (at 68 per cent confidence limit), of the FRB luminosity function assuming cosmic FRB rate of 104 FRBs per sky per day with flux above 1 Jy located out to redshift of 1. Further FRB surveys of galaxy clusters with high-sensitivity instruments will tighten the constraints on the faint end of the luminosity function and, thus, are strongly encouraged.