  The main contribution of this article is the construction of a finite time
blow-up solution to the mass-critical focusing nonlinear Schr\"odinger equation
set on a metric star graph $\mathcal G$ with $N$ edges, for any $N\ge 2$. After
establishing well-posedness of the corresponding Cauchy problem in
$H^1(\mathcal G)$, we obtain the sharp threshold for global existence in terms
of the mass of the ground state, called minimal mass. We then construct a
minimal mass solution which blows up in finite time at the vertex of $\mathcal
G$. The blow-up profile and blow-up speed are characterized explicitly.
