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In this paper we consider the following problem: Over the class of all simpleconnected unicyclic graphs on $n$ vertices with girth $g$ ($n,g$ being fixed),which graph minimizes the Laplacian spectral radius? We prove that the graph$U_{n,g}$ (defined in Section 1) uniquely minimizes the Laplacian spectralradius for $n\geq 2g-1$ when $g$ is even and for $n\geq 3g-1$ when $g$ is odd.

Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth