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We show that a widely believed conjecture concerning rigidity of genus-zero and-one holomor-phic curves in Calabi-Yau threefolds implies a relation between the genus-one GW-invariants of a quintic threefold in P4 and the genus-zero and genus-one GW-invariants of P4. This relation is a special case of a general formula for the genus-one GW-invariants of complete intersections obtained in a previous paper. In contrast to the general case, this paper's derivation is more ge-ometric and makes direct use of the rigidity property. Thus, it provides further evidence for the rigidity conjecture in low genera. On the other hand, this paper also suggests a potential way of disapproving the less commonly believed generalization of the rigidity conjecture to arbitrar