  We investigate a Born--Infeld-type model of nonlinear electrodynamics,
possessing three parameters, coupled with general relativity. As a particular
case Born--Infeld electrodynamics is reproduced. There is no singularity of the
electric field at the centre of point-like charged particles and self-energy of
charges is finite in this model. The magnetized black hole is studied and
solutions are obtained. We demonstrate that for some parameters of the model
the black hole is regular. We find the asymptotic of the metric and mass
functions at $r\rightarrow\infty$ and $r\rightarrow 0$, and corrections to the
Reissner--Nordstr\"{o}m solution. Thermodynamics of black holes is
investigated. We calculate the Hawking temperature of black holes and show that
black holes are not stable and there are phase transitions in the model under
consideration.
