  The COMPASS experiment recently discovered a new isovector resonance-like
signal with axial-vector quantum numbers, the $a_1(1420)$, decaying to
$f_0(980)\pi$. With a mass too close to and a width smaller than the
axial-vector ground state $a_1(1260)$, it was immediately interpreted as a new
light exotic meson, similar to the $X$, $Y$, $Z$ states in the hidden-charm
sector. We show that a resonance-like signal fully matching the experimental
data is produced by the decay of the $a_1(1260)$ resonance into $K^\ast(\to
K\pi)\bar{K}$ and subsequent rescattering through a triangle singularity into
the coupled $f_0(980)\pi$ channel. The amplitude for this process is calculated
using a new approach based on dispersion relations. The triangle-singularity
model is fitted to the partial-wave data of the COMPASS experiment. Despite
having less parameters, this fit shows a slightly better quality than the one
using a resonance hypothesis and thus eliminates the need for an additional
resonance in order to describe the data. We thereby demonstrate for the first
time in the light-meson sector that a resonance-like structure in the
experimental data can be described by rescattering through a triangle
singularity, providing evidence for a genuine three-body effect.
