Thin films of Ba1-x SrxTiO3 (BST) are being actively investigated for applications in dynamic random access memories (DRAM) because of their properties such as high dielectric constant, low leakage current, and low fatigue. Several approaches have been used to improve the properties of thin films such as doping with aliovalent dopants, graded compositions, and layered structures. We have found that interposing layers of an electronic insulator such as ZrO2 in between BST layers results in a significant reduction in the leakage current. In this paper the low temperature electrical properties of these multilayer structures are reported. The structures consist of alternate layers of Ba0.8Sr0.2TiO3 and ZrO2 deposited by a sol-gel process on platinized Si substrates. The thickness and the number of layers are varied while keeping the total thickness of the film constant. Multiple peaks in the dielectric constant vs temperature plots at all frequencies are observed in the multilayered films. The properties of the multilayer films are a complex function of the number of layers and their thicknesses. A structure with several thin layers of ZrO2 interposed between the BST layers produces smoother plots than a single layer of ZrO2 of same total thickness. This is attributed to more uniform distribution, as determined by XPS, of ZrO2 in the multilayer structure due to smaller diffusion distances.