Conversing on cell phones while driving an automobile is a common practice. We examine the interference of the cognitive load of conversational dialog with driving tasks, with the goal of identifying better and worse times for conversations during driving. We present results from a controlled study involving 18 users using a driving simulator. The driving complexity and conversation type were manipulated in the study, and performance was measured for factors related to both the primary driving task and secondary conversation task. Results showed significant interactions between the primary and secondary tasks, where certain combinations of complexity and conversations were found especially detrimental to driving. We present the studies and analyses and relate the findings to prior work on multiple resource models of cognition. We discuss how the results can frame thinking about policies and technologies aimed at enhancing driving safety.