  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 Kilometer Array Pathfinder telescope. During 300 hr 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, $\alpha<1.52$ (at 68\%
confidence limit), and the minimum luminosity, $L_{\rm min}\gtrsim 2\times
10^{40}$ erg s$^{-1}$ (at 68\% confidence limit), of the FRB luminosity
function assuming cosmic FRB rate of $10^4$ FRBs sky$^{-1}$ day$^{-1}$ 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.
