  The $2$-star model is the simplest exponential random graph model that
displays complex behavior, such as degeneracy and phase transition. Despite its
importance, this model has been solved only in the regime of dense
connectivity. In this work we solve the model in the finite connectivity
regime, far more prevalent in real world networks. We show that the model
undergoes a condensation transition from a liquid to a condensate phase along
the critical line corresponding, in the ensemble parameters space, to the
Erd\"os-R\'enyi graphs. In the fluid phase the model can produce graphs with a
narrow degree statistics, ranging from regular to Erd\"os-R\'enyi graphs, while
in the condensed phase, the "excess" degree heterogeneity condenses on a single
site with degree $\sim\sqrt{N}$. This shows the unsuitability of the two-star
model, in its standard definition, to produce arbitrary finitely connected
graphs with degree heterogeneity higher than Erd\"os-R\'enyi graphs and
suggests that non-pathological variants of this model may be attained by
softening the global constraint on the two-stars, while keeping the number of
links hardly constrained.
