In this paper, we report direct evidence of a structural transition in the organic superconductor $\ensuremath{\kappa}\ensuremath{-}{(\mathrm{B}\mathrm{E}\mathrm{D}\mathrm{T}\ensuremath{-}\mathrm{T}\mathrm{T}\mathrm{F})}_{2}{\mathrm{Cu}[\mathrm{N}(\mathrm{CN})}_{2}]\mathrm{Br}$ near 80 K and the effect of disorder on the superconducting transition temperature. By cooling the sample from above 80 K, the interlayer magnetoresistance displays a bumplike feature, which increases sharply with increasing cooling rate. The rapidly cooled sample has a much larger resistivity and a lower transition temperature, which decreases linearly with increasing resistivity near the transition temperature. We propose that rapid cooling quenches the sample into a disordered state. Localized moments in the disordered state reduce the superconducting transition temperature.