In this work we will analyze branching Brownian motion on a finite interval with one absorbing and one reflecting boundary, having constant drift rate toward the absorbing boundary. Similar processes have been considered by Kesten, and more recently by Harris, Hesse, and Kyprianou. The current offering is motivated largely by the utility of such processes in modeling a biological population's response to climate change. We will demonstrate a critical parameter value that answers the fundamental question of whether persistence is possible for our branching process, or if extinction is inevitable. The bulk of the work is done by the distinguished path (or ``spine") analysis for branching processes.