  We compute the Chow rings with integral coefficients of moduli stacks of
minimal Weierstrass fibrations over the projective line. For each integer
$N\geq 1$, there is a moduli stack $\mathcal{W}^{\mathrm{min}}_N$ parametrizing
minimal Weierstrass fibrations with fundamental invariant $N$. Following work
of Miranda and Park--Schmitt, we give a quotient stack presentation for each
$\mathcal{W}^{\mathrm{min}}_N$. Using these presentations and equivariant
intersection theory, we determine a complete set of generators and relations
for each of the Chow rings. For the cases $N=1$ (respectively, $N=2$),
parametrizing rational (respectively, K3) elliptic surfaces, we give a more
explicit computation of the relations.
