  An exponential model of nonlinear electrodynamics coupled with the
gravitational field is analyzed. It was shown that weak, dominant, and strong
energy conditions are satisfied. We obtain corrections to Coulomb's law at
$r\rightarrow\infty$. It was demonstrated that electric-magnetic duality
between ${\cal F}$ and ${\cal P}$ frames is violated. The asymptotic black hole
solution at $r\rightarrow \infty$ is found that contains corrections to the
Reissner-Nordstr\"om solution. The electromagnetic mass of the black hole is
calculated, expressed through the parameter $\beta$ of the model.
