Acousto-electric tomography (AET) involves three steps to retrieve the distribution of conductivity in a domain of interest (DOI): measure the potential on the DOI boundary (usually with a limited number of electrodes set there), compute the power density from this potential, use it to retrieve the distribution sought after. Almost all developed algorithms for AET assume that the power density is known, so their focus is mostly on the last step. A complete framework for AET is proposed herein to connect the three steps, the complete electrode model (CEM) being used to simulate the voltages measured on electrodes. The potential on the whole DOI boundary is reconstructed from such voltages. Then, the power density is computed, and the conductivity distribution in the DOI is retrieved. A method based on singular value decomposition (SVD) is proposed. This method and the iterative Levenberg-Marquardt method are used for numerical illustration. The SVD-based method yields the potential on the whole DOI boundary, and a gross map of the conductivity distribution is also obtained, to serve as an initial guess of the Levenberg-Marquardt method to yield the conductivity contribution with higher accuracy.