Many viruses express factors that reduce host gene expression through widespread degradation of cellular mRNA. An example of this class of proteins is the mRNA-targeting endoribonuclease SOX from the gamma-herpesvirus Kaposi?s sarcoma-associated herpesvirus (KSHV). Previous studies indicated that cleavage of messenger RNAs (mRNA) by SOX occurs at specific locations defined by the sequence of the target RNA, which is at odds with the down-regulation of a large portion of cellular transcripts. In this study, we address this paradox by using high-throughput sequencing of cleavage intermediates combined with a custom bioinformatics-based analysis pipeline to identify SOX cleavage sites across the mRNA transcriptome. These data, coupled with targeted mutagenesis, reveal that while cleavage sites are specific and reproducible, they are defined by a degenerate sequence motif containing a small number of conserved residues rather than a strong consensus sequence. This degenerate element is well represented in both human and KSHV mRNA, and its presence correlates with RNA destabilization by SOX. This represents a new endonuclease targeting strategy, in which use of a degenerate targeting element enables RNA cleavage at specific locations without restricting the range of targets. Furthermore, it shows that strong target selectivity can be achieved without a high degree of sequence specificity.