We have measured the effects of dilute magnetic-atom doping on the superconducting transition temperature of tungsten thin films. Our “Tc tuning” technique is accurate, precise, and simple. Experiments were performed using dc-magnetron-sputtered tungsten films with undoped values of Tc in the range of 70–150 mK. The magnetic-atom doping was achieved using ion implantation. Specific Tc suppressions of between 5% and 65% were targeted and observed in this study. The transition width of each undoped sample was ≈1 mK and the transition widths remained sharp after implantation with 56Fe+ ions. Our data are in good agreement with predictions of a linear dependence of Tc suppression with increasing magnetic-atom concentration, in the small concentration limit. At higher concentrations, antiferromagnetic coupling between the magnetic dopant atoms becomes important and the Tc-suppression effect is diminished. We use our Tc data to calculate the Abrikosov–Gor’kov (AG) and Ruderman–Kittel–Kasuya–Yosida (RKKY) spin–flip relaxation parameters τAG and τRKKY. We conclude with a brief discussion of applications of the Tc-tuning technique, and present our plans for future studies in this area.