We present the first observational hints of the severe removal of both molecular and H i gas from the dwarf galaxy VCC 1249. This extreme stripping event is thought to be driven by the combined effects of tidal interaction and ram pressure. Using deep CO (2−1) observations from the James Clerk Maxwell Telescope, we obtained marginal CO detections in three regions within the stripped H i tail, with molecular masses of ∼10⁵–10⁶ M_⊙, comparable to typical masses of giant molecular clouds. In contrast, we did not find CO emission within the stellar disk of VCC 1249. This indicates the severe removal of cold gas, which likely caused the sudden cessation of star formation in the galaxy. This identifies VCC 1249 as a unique laboratory for witnessing the rapid, environmentally driven quenching of a dwarf galaxy. Our findings provide a critical observational link between gas removal mechanisms and the dramatic phase transition of cluster dwarfs from star-forming to quiescent systems.