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 $N \geq 2$. 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.