How biological systems such as proteins achieve robustness to ubiquitous perturbations is a fundamental biological question. Such perturbations include errors that introduce phenotypic mutations into nascent proteins during the translation of mRNA. These errors are remarkably frequent. They are also costly, because they reduce protein stability and help create toxic misfolded proteins. Adaptive evolution might reduce these costs of protein mistranslation by two principal mechanisms. The first increases the accuracy of translation via synonymous "high fidelity" codons at especially sensitive sites. The second increases the robustness of proteins to phenotypic errors via amino acids that increase protein stability. To study how these mechanisms are exploited by populations evolving in the laboratory, we evolved the antibiotic resistance gene TEM-1 in Escherichia coli hosts with either normal or high rates of mistranslation. We analyzed TEM-1 populations that evolved under relaxed and stringent selection for antibiotic resistance by single molecule real-time sequencing. Under relaxed selection, mistranslating populations reduce mistranslation costs by reducing TEM-1 expression. Under stringent selection, they efficiently purge destabilizing amino acid changes. More importantly, they accumulate stabilizing amino acid changes rather than synonymous changes that increase translational accuracy. In the large populations we study, and on short evolutionary timescales, the path of least resistance in TEM-1 evolution consists of reducing the consequences of translation errors rather than the errors themselves.