We demonstrate here for the first time that the liquid phase deposition (LPD) method, a simple, low cost and highly reproducible synthetic approach generally used for the deposition of high quality metal oxide thin films, can be reliably extended to the rational design of 1D magnetoelectric core-shell nano-architectures. In the first step of the process, highly crystalline ferroelectric (BaTiO3) nanotubes with an average diameter of 200 nm and controllable wall thickness were synthesized by the controlled hydrolysis of metal oxyfluoride precursors upon immersing alumina templates into a treatment solution at temperatures as low as 40 oC. The resulting perovskite nanotubes immobilized within the channels of the anodic aluminum oxide (AAO) membranes have been subsequently filled with a spinel ferrite phase, with the chemical composition Zn1.5Fe1.5O4 yielding spinel-perovskite 1D core-shell magnetoelectric architectures. The resulting core-shell tubular nanocomposites have been characterized structurally, morphologically and compositionally and their ferroelectric, magnetic and magnetoelectric properties have been measured. A change from a superparamagnetic to a ferrimagnetic behavior was observed when the pristine spinel ferrite nanotubes were incorporated into the spinel-perovskite core-shell nanocomposites, indicating the existence of a magnetoelectric coupling between the two ferroic phases. Moreover, the measured magnetoelectric coupling coefficient was α=1.08 V/cm·Oe, value which is superior to the values reported for similar thin film and tubular spinel ferrite magnetoelectric nanocomposites, thereby indicating a strong strain-mediated coupling between the ferroelectric and magnetostrictive phase in the 1D core-shell nanocomposites and making these materials suitable for implementation into various functional devices.