Since Bennett's 1973 seminal paper, there has been a growing interest in general-purpose, reversible computations and they have been studied using both mathematical and physical models. Following Bennett, given a terminating computation of a deterministic Turing Machine, one may be interested in constructing a new Turing Machine, whose computation consists of two stages. The first stage emulates the original Turing Machine computation on its working tape, while also producing the trace of the computation on a new history tape. The second stage reverses the first stage using the trace information. Ideally, one would want the second stage to traverse whole-machine states in the reverse order from that traversed in the first stage. But this is impossible other than for trivial computations. Bennett constructs the second phase by using additional controller states, beyond those used during the first stage. In this report, a construction of the new machine is presented in which the second stage uses the same and only those controller states that the first stage used and they are traversed in the reverse order. The sole element that is not fully reversed is the position of the head on the history tape, where it is out of phase by one square compared to the first stage.