The 3' exonucleolytic processing of stable RNAs is conserved throughout biology. Yeast strains lacking the exoribonuclease Rex1 are defective in the 3' processing of stable RNAs, including 5S rRNA and tRNA. The equivalent RNA processing steps in  Escherichia coli  are carried out by RNase T. Rex1 is larger than RNase T, the catalytic DEDD domain being embedded within uncharacterised N- and C-terminal regions. Here we report that both N- and C-terminal regions of Rex1 are essential for its function, as shown by genetic analyses and 5S rRNA profiling. Full-length Rex1, but not mutants lacking N- or C-terminal regions, accurately processed a 3' extended 5S rRNA substrate. Crosslinking analyses showed that both N- and C-terminal regions of Rex1 directly contact RNA  in vivo  Sequence homology searches identified YFE9 in  Schizosaccharomyces pombe  and SDN5 in  Arabidopsis thaliana  as closely related proteins to Rex1. In addition to the DEDD domain, these proteins share a domain, referred to as the RYS (Rex1, YFE9 and SDN5) domain, that includes elements of both the N- and C-terminal flanking regions. We also characterise a nuclear localisation signal in the N-terminal region of Rex1. These studies reveal a novel dual domain structure at the core of Rex1-related ribonucleases, wherein the catalytic DEDD domain and the RYS domain are aligned such that they both contact the bound substrate. The domain organisation of Rex1 is distinct from that of other previously characterised DEDD family nucleases and expands the known repertoire of structures for this fundamental family of RNA processing enzymes.