  Using lattice simulations we calculate the rate of baryon number violating
processes, the sphaleron rate, in the Standard Model with an external
(hyper)magnetic field for temperatures across the electroweak cross-over,
focusing on the broken phase. Additionally, we compute the Higgs expectation
value and the pseudocritical temperature. The electroweak cross-over shifts to
lower temperatures with increasing external magnetic field, bringing the onset
of the suppression of the baryon number violation with it. When the
hypermagnetic field reaches the magitude $B_Y \approx 2 T^2$ the cross-over
temperature is reduced from $160$ GeV to $145$ GeV. In the broken phase for
small magnetic fields the rate behaves quadratically as a function of the
magnetic flux. For stronger magnetic fields the rate reaches a linear regime
which lasts until the field gets strong enough to restore the electroweak
symmetry where the symmetric phase rate is reached.

High Energy Physics - Phenomenology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Lattice