  We consider the possibility that the dark sector of our Universe contains a
negative cosmological constant dubbed $\lambda$. For such models to be viable,
the dark sector should contain an additional component responsible for the
late-time accelerated expansion rate ($X$). We explore the departure of the
expansion history of these models from the concordance $\Lambda$ Cold Dark
Matter model. For a large class of our models the accelerated expansion is
transient with a nontrivial dependence on the model parameters. All models with
$w_X>-1$ will eventually contract and we derive an analytical expression for
the scale factor $a(t)$ in the neighborhood of its maximal value. We find also
the scale factor for models ending in a Big Rip in the regime where dustlike
matter density is negligible compared to $\lambda$. We address further the
viability of such models, in particular when a high $H_0$ is taken into
account. While we find no decisive evidence for a nonzero $\lambda$, the best
models are obtained with a phantom behavior on redshifts $z\gtrsim 1$ with a
higher evidence for nonzero $\lambda$. An observed value for $h$ substantially
higher than $0.70$ would be a decisive test of their viability.

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology