The tRNA m1A58 methyltransferase is composed of two subunits encoded by the essential genes TRM6 and TRM61 (formerly GCD10 and GCD14). The trm6-504 mutation results in a defective m1A methyltransferase (Mtase) and a temperature-sensitive growth phenotype that is attributable to the absence of m1A58 and consequential tRNAiMet instability. We used a genetic approach to identify the genes responsible for tRNAiMet degradation in trm6 cells. Three recessive extragenic mutations that suppress trm6-504 mutant phenotypes and restore hypomodified tRNAiMet to near normal levels were identified. The wild-type allele of one suppressor, DIS3/RRP44, encodes a 3′-5′ exoribonuclease and a member of the multisubunit exosome complex. We provide evidence that a functional nuclear exosome is required for the degradation of tRNAiMet lacking m1A58. A second suppressor gene encodes Trf4p, a DNA polymerase (pol σ) with poly(A) polymerase activity. Whereas deletion of TRF4 leads to stabilization of tRNAiMet, overexpression of Trf4p destabilizes the hypomodified tRNAiMet in trm6 cells. The hypomodified, but not wild-type, pre-tRNAiMet accumulates as a polyadenylated species, whose abundance and length distribution both increase upon Trf4p overexpression. These data indicate that a tRNA surveillance pathway exists in yeast that requires Trf4p and the exosome for polyadenylation and degradation of hypomodified pre-tRNAiMet.