An exercise of Hartshorne shows that, given $A= k[x,y,z]/(xy-z^2)$, $\mathrm{Cl}(A)=\mathbb Z/2\mathbb Z$.
I must prove that $\mathrm{Pic}(\mathrm{Spec}\ A)=0$, thus giving a counter-example to $\mathrm{Pic}=\mathrm{Cl}$.
My attempt was to consider some UFD $B$, e.g. $k[x,y,z]$ itself, send it to $A$, and obtain a map from $Pic(\rm Spec\ B)$ to $Pic(\rm Spec\ A)$, and then hope that this is surjective. This would conclude since $B$ UFD implies $Pic(\rm Spec\ B)=0$.
Can this work? Any hints are welcome.