Alcubierre recently exhibited a spacetime which, within the framework of general relativity, allows travel at superluminal speeds if matter with a negative energy density can exist, and conjectured that it should be possible to use similar techniques to construct a theory containing closed causal loops and, thus, travel backwards in time. We verify this conjecture by exhibiting a simple modification of Alcubierre's model, requiring no additional assumptions, in which causal loops are possible. We also note that this mechanism for generating causal loops differs in essential ways from that discovered by Gott involving cosmic strings. \textcopyright{} 1996 The American Physical Society.