  In the present paper the small Quantum Cohomology ring of some Fano
threefolds which are obtained as one- or two-curve blow-ups from $P^3$ or the
quadric $Q^3$ is explicitely computed. Because of systematic usage of the
associativity property of quantum product only a very small and enumerative
subset of Gromov-Witten invariants is needed. Then, for these threefolds the
Dubrovin conjecture on the semisimplicity of Quantum Cohomology is proven by
checking the computed Quantum Cohomology rings and by showing that a smooth
Fano threefold $X$ with $b_3(X)=0$ admits a complete exceptional set of the
appropriate length.
