We prove the modularity of mixed periods associated with conifold fibres of families of Calabi-Yau threefolds. This is done by "fibering out", i.e. by expressing these periods as integrals of periods of families of K3 surfaces and by using modularity properties of the latter. Besides classical periods of holomorphic modular forms and meromorphic modular forms with vanishing residues, the computations lead to new interesting periods associated with meromorphic modular forms with non-vanishing residues as well as contours between CM points. We use these period values establish in the conifold fibre and mirror symmetry to establish exact large degree asymptotics of enumerative invariants on the mirror Calabi-Yau threefold.
