We report the first observation of a transient relativistic jet from the canonical black hole candidate, Cygnus X-1, obtained with the Multi-Element Radio-Linked Interferometer Network (MERLIN). The jet was observed in only one of six epochs of MERLIN imaging of the source during a phase of repeated X-ray spectral transitions in 2004 Jan--Feb, and this epoch corresponded to the softest 1.5-12 keV X-ray spectrum. With only a single epoch revealing the jet, we cannot formally constrain its velocity. Nevertheless, several lines of reasoning suggest that the jet was probably launched 0.5-4.0 days before this brightening, corresponding to projected velocities of 0.2c < v_app < 1.6c, and an intrinsic velocity of > 0.3c. We also report the occurrence of a major radio flare from Cyg X-1, reaching a flux density of ~120 mJy at 15 GHz, and yet not associated with any resolvable radio emission, despite a concerted effort with MERLIN. We discuss the resolved jet in terms of the recently proposed unified model for the disc-jet coupling in black hole X-ray binaries, and tentatively identify the jet line for Cyg X-1. The source is consistent with the model in the sense that a steady jet appears to persist initially when the X-ray spectrum starts softening, and that once the spectral softening is complete the core radio emission is suppressed and transient ejecta / shock observed. However, there are some anomalies, and Cyg X-1 clearly does not behave like a normal black hole transient in progressing to the canonical soft / thermal state once the ejection event has happened.
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