  We provide an analytical solution to the quantum dynamics of a flat
Friedmann-Lema\^itre- Robertson-Walker model with a massless scalar field in
the presence of a small and positive cosmological constant, in the context of
Loop Quantum Cosmology. We use a perturbative treatment with respect to the
model without a cosmological constant, which is exactly solvable. Our solution
is approximate, but it is precisely valid at the high curvature regime where
quantum gravity corrections are important. We compute explicitly the evolution
of the expectation value of the volume. For semiclassical states characterized
by a Gaussian spectral profile, the introduction of a positive cosmological
constant displaces the bounce of the solvable model to lower volumes and to
higher values of the scalar field. These displacements are state dependent, and
in particular, they depend on the peak of the Gaussian profile, which measures
the momentum of the scalar field. Moreover, for those semiclassical states, the
bounce remains symmetric, as in the vanishing cosmological constant case.
However, we show that the behavior of the volume is more intricate for generic
states, leading in general to a non-symmetric bounce.
