U6 spliceosomal RNA has a complex secondary structure that includes a highly conserved stemloop near the 3' end. The 3' stem is unwound when U6 RNA base-pairs with U4 RNA during spliceosome assembly, but likely reforms when U4 RNA leaves the spliceosome prior to the catalysis of splicing. A mutation in yeast U6 RNA that hyperstabilizes the 3' stem confers cold sensitivity and inhibits U4/U6 assembly as well as a later step in splicing. Here we show that extragenic suppressors of the 3' stem mutation map to the gene coding for splicing factor Prp24. The suppressor mutations are located in the second and third of three RNA-recognition motifs (RRMs) in Prp24 and are predicted to disrupt RNA binding. Mutations in U6 RNA predicted to destabilize a novel helix adjacent to the 3' stem also suppress the 3' stem mutation and enhance the growth defect of a suppressor mutation in RRM2 of Prp24. Both phenotypes are reverted by a compensatory mutation that restores pairing in the novel helix. These results are best explained by a model in which RRMs 2 and 3 of Prp24 stabilize an extended intramolecular structure in U6 RNA that competes with the U4/U6 RNA interaction, and thus influence both association and dissociation of U4 and U6 RNAs during the splicing cycle.