Lethal mutagenesis is a transition towards virus extinction mediated by enhanced mutation rates during viral genome replication. Theoretical studies suggest that viruses can evolve towards regions of their fitness landscapes at which they display resistance to the deleterious effects of mutations. It has been suggested that such mutational robustness could jeopardize lethal mutagenesis. We have used the Arenavirus lymphocytic choriomeningitis virus (LCMV) to explore whether treatment with the mutagenic base analogue 5-fluorouracil (FU) selected for viral populations displaying resistance to lethal mutagenesis. Neither average LCMV populations with a history of FU mutagenesis, nor individual biological LCMV clones derived from those populations, displayed any resistance to lethal mutagenesis by FU. They were as sensitive to FU-induced extinction as LCMV populations and clones treated in parallel, but without a history of FU mutagenesis. Current evidence of the molecular events affecting quasispecies dynamics suggests that it is unlikely that a viral population can acquire mutational robustness under the increased mutation rates associated with mutagenic treatments. We consider mechanisms by which viruses could escape extinction by lethal mutagenesis, and provide evidence that mutational robustness is unlikely to be one of them.