  Let $C$ be a smooth projective curve over the field of complex numbers
$\mathbb{C}$ of genus $g(C)>0$. Let $E$ be a locally free sheaf on $C$ of rank
$r$ and degree $e$. Let $\mathcal{Q}:={\rm Quot}_{C/\mathbb{C}}(E,k,d)$ denote
the Quot scheme of quotients of $E$ of rank $k$ and degree $d$. For $k>0$ and
$d\gg 0$ we compute the Picard group of $\mathcal{Q}$.
