  Starting from the most general action in Einstein-Dilaton-Nonlinear
Electrodynamics (NED) theory, we obtain the field equations. We apply the field
equations for the specific NED known as the Liouville type plus a cosmological
constant and solve the field equations. With a pure magnetic field it is shown
that the square root model is the strong field limit of the Born-Infeld NED
theory. In static spherically symmetric spacetime and a magnetic monopole
sitting at the origin, the field equations are exactly solvable provided the
intergation constants of the solution and the theory constants appearing in the
action are linked through two constraints. As it is known such an exact
solution in the absence of dilaton i.e., gravity coupled to square root NED
doesn't exist. Therefore, the presence of the dilaton gives additional freedom
to solve the field equations. The obtained spacetime is singular and
non-asymptotically flat and depending on the free parameters it may be a black
hole or a cosmological object. For the black hole spacetime, we study the
thermal stability of the spacetime and show that the black hole is thermally
stable provided its size is larger than a critical value.
