The small nuclear RNA (snRNA) components of the spliceosome undergo many conformational rearrangements during its assembly, catalytic activation and disassembly. The U4 and U6 snRNAs are incorporated into the spliceosome as a base-paired complex within the U4/U6.U5 small nuclear ribonucleoprotein (tri-snRNP). U4 and U6 are then unwound in order for U6 to pair with U2 to form the spliceosome's active site. After splicing, U2/U6 is unwound and U6 annealed to U4 to reassemble the tri-snRNP. U6 rearrangements are crucial for spliceosome formation but are poorly understood. We have used single-molecule Förster resonance energy transfer and unwinding assays to identify interactions that promote U4/U6 unwinding and have studied their impact in yeast. We find that U4/U6 is efficiently unwound using DNA oligonucleotides by coupling unwinding of U4/U6 stem II with strand invasion of stem I. Unwinding is stimulated by the U6 telestem, which transiently forms in the intact U4/U6 RNA complex. Stabilization of the telestem in vivo results in accumulation of U4/U6 di-snRNP and impairs yeast growth. Our data reveal conserved mechanisms for U4/U6 unwinding and indicate telestem dynamics are critical for tri-snRNP assembly and stability.