We present a direct measurement of the quenching of nonlocal heat transport in a laser-produced plasma by applying large external magnetic fields (>10 T). The temporally resolved Thomson-scattering measurements of the electron temperature profile show that the heat front propagation transverse to a high-power laser beam is slowed resulting in extremely strong local heating. We find agreement with hydrodynamic modeling when including a magnetic field model that self-consistently evolves the fields in the plasma.