Let $R$ be a finitely generated commutative ring over a field, for concreteness.

If $S,T \leq R$ are two finitely generated subrings, is their intersection also finitely generated?

(Certainly this isn't true for infinite intersections.)

If so,

How can one compute such an intersection using a computer algebra package?

It's easy to reduce to the case that $R = {\mathrm k}[s_1,\ldots,s_n,t_1,\ldots,t_m]/I$, the subring $S$ is generated by $s_1,\ldots,s_n$, and $T$ is generated by $t_1,\ldots,t_m$.