The main aim of this paper is to give a description for the structure of the (small) quantum cohomology ring of the toric variety $X={\bb P}(\oplus _{i=1}^r{\cal O}_{{\bb P}^1}(a_i))$ with $\sum_{i=1}^ra_i=\epsilon +k r$ and $\epsilon \in \{0,1 \}$. As we explain later, the importance of this result relies on the fact that, unless $k =0$, $X$ is a non-Fano toric variety and the fact that we determine not only a presentation of the quantum cohomology ring $QH^*(X;{\bb Z})$ but also all quantum products $\alpha* \beta$ with $\alpha, \beta \in H^*(X;{\bb Z})$ or, equivalently, all three-point genus-0 Gromov–Witten invariants.