The suppression of superconductivity by nonmagnetic disorder is investigated systematically in the organic superconductor $kappa$-(BEDT-TTF)$_2$Cu(NCS)$_2$. We introduce a nonmagnetic disorder arising from molecule substitution in part with deuterated BEDT-TTF or BMDT-TTF for BEDT-TTF molecules and molecular defects introduced by X-ray irradiation. A quantitative evaluation of the scattering time $tau_{rm dHvA}$ is carried out by de Haas-van Alphen (dHvA) effect measurement. A large reduction in $T_{rm c}$ with a linear dependence on $1/tau_{rm dHvA}$ is found in the small-disorder region below $1/tau_{rm dHvA} simeq$ 1 $times$ 10$^{12}$ s$^{-1}$ in both the BMDT-TTF molecule-substituted and X-ray-irradiated samples. The observed linear relation between $T_{rm c}$ and $1/tau_{rm dHvA}$ is in agreement with the Abrikosov-Gorkov (AG) formula, at least in the small-disorder region. This observation is reasonably consistent with the unconventional superconductivity proposed thus far for the present organic superconductor. A deviation from the AG formula, however, is observed in the large-disorder region above $1/tau_{rm dHvA} simeq$ 1 $times$ 10$^{12}$ s$^{-1}$, which reproduces the previous transport study (J. G. Analytis {it et al.}: Phys. Rev. Lett. {bf 96} (2006) 177002). We present some interpretations of this deviation from the viewpoints of superconductivity and the inherent difficulties in the evaluation of scattering time.
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