  A heat flux in a high-$\beta$ plasma with low collisionality triggers the
whistler instability. Quasilinear theory predicts saturation of the instability
in a marginal state characterized by a heat flux that is fully controlled by
electron scattering off magnetic perturbations. This marginal heat flux does
not depend on the temperature gradient and scales as $1/\beta$. We confirm this
theoretical prediction by performing numerical particle-in-cell simulations of
the instability. We further calculate the saturation level of magnetic
perturbations and the electron scattering rate as functions of $\beta$ and the
temperature gradient to identify the saturation mechanism as quasilinear.
Suppression of the heat flux is caused by oblique whistlers with
magnetic-energy density distributed over a wide range of propagation angles.
This result can be applied to high-$\beta$ astrophysical plasmas, such as the
intracluster medium, where thermal conduction at sharp temperature gradients
along magnetic-field lines can be significantly suppressed. We provide a
convenient expression for the amount of suppression of the heat flux relative
to the classical Spitzer value as a function of the temperature gradient and
$\beta$. For a turbulent plasma, the additional independent suppression by the
mirror instability is capable of producing large total suppression factors
(several tens in galaxy clusters) in regions with strong temperature gradients.
