We investigate the lattice structure of heterostructures comprising of ferroelectric BaTiO$_3$ (BTO) thin films and BaBiO$_3$ (BBO), the insulating parent-compound of the high Tc superconductor. Motivated by theoretical predictions of exotic phenomena in BBO-based heterostructures including interfacial conductivity, superconductivity and topologically-protected states, we synthesize BTO/BBO heterostructures by molecular beam epitaxy and characterize their structural properties by in-situ reflection high energy electron diffraction and ex-situ high-resolution synchrotron X-ray diffraction, Raman spectroscopy and piezoforce microscopy. For heterostructures with 4 uc BBO layers, reciprocal space maps indicate strain relaxation relative to the SrTiO$_3$ substrate. We observe a strong tetragonal distortion for heterostructures with 2 unit cells BBO coherently strained to the BTO layers. Raman spectroscopy measurements indicate a suppression of the breathing mode distortion associated with the charge density wave in bulk BBO. The ferroelectric properties of the system are confirmed by piezoforce microscopy measurements. The coupling between the ferroelectric polarization and the electronic states in BBO may potentially serve as a starting point for tunable electrostatic doping to realize the novel predicted states in the atomically-thin BBO layers.