  In this work, a rigorous proof of the orbital stability of the black soliton
solution of the quintic Gross-Pitaevskii equation in one spatial dimension is
obtained. We first build and show explicitly black and dark soliton solutions
and we prove that the corresponding Ginzburg-Landau energy is coercive around
them by using some orthogonality conditions related to perturbations of the
black and dark solitons. The existence of suitable perturbations around black
and dark solitons satisfying the required orthogonality conditions is deduced
from an Implicit Function Theorem. In fact, these perturbations involve dark
solitons with sufficiently small speeds and some proportionality factors
ari\-sing from the explicit expression of their spatial derivative. We are also
able to control the evolution of the modulation parameters along the quintic
Gross-Pitaevskii flow by estimating their growth in time. As a direct
consequence, we also prove the orbital stability of the dark soliton in a small
speed interval.
