We investigate self-adjoint extensions of the minimal Kirchhoff Laplacian on an infinite metric graph. More specifically, the main focus is on the relationship between graph ends and the space of self-adjoint extensions of the corresponding minimal Kirchhoff Laplacian $mathbf{H}_0$. First, we introduce the notion of finite and infinite volume for (topological) ends of a metric graph and then establish a lower bound on the deficiency indices of $mathbf{H}_0$ in terms of the number of finite volume graph ends. This estimate is sharp and we also find a necessary and sufficient condition for the equality between the number of finite volume graph ends and the deficiency indices of $mathbf{H}_0$ to hold. Moreover, it turns out that finite volume graph ends play a crucial role in the study of Markovian extensions of $mathbf{H}_0$. In particular, we show that the minimal Kirchhoff Laplacian admits a unique Markovian extension exactly when every topological end of the underlying metric graph has infinite volume. In the case of finitely many finite volume ends (for instance, the latter includes Cayley graphs of a large class of finitely generated infinite groups) we are even able to provide a complete description of all Markovian extensions of $mathbf{H}_0$.
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