We report the detection of a strong, organized magnetic field in the secondary component of the massive O8III/I+O7.5V/III double-lined spectroscopic binary system HD 47129 (Plasketts star), in the context of the Magnetism in Massive Stars (MiMeS) survey. Eight independent Stokes $V$ observations were acquired using the ESPaDOnS spectropolarimeter at the Canada-France-Hawaii Telescope and the Narval spectropolarimeter at the Telescope Bernard Lyot. Using Least-Squares Deconvolution we obtain definite detections of signal in Stokes $V$ in 3 observations. No significant signal is detected in the diagnostic null ($N$) spectra. The Zeeman signatures are broad and track the radial velocity of the secondary component; we therefore conclude that the rapidly-rotating secondary component is the magnetized star. Correcting the polarized spectra for the line and continuum of the (sharp-lined) primary, we measured the longitudinal magnetic field from each observation. The longitudinal field of the secondary is variable and exhibits extreme values of $-810pm 150$ G and $+680pm 190$ G, implying a minimum surface dipole polar strength of $2850pm 500$ G. In contrast, we derive an upper limit ($3sigma$) to the primarys surface magnetic field of 230 G. The combination of a strong magnetic field and rapid rotation leads us to conclude that the secondary hosts a centrifugal magnetosphere fed through a magnetically confined wind. We revisit the properties of the optical line profiles and X-ray emission - previously interpreted as a consequence of colliding stellar winds - in this context. We conclude that HD 47129 represents a heretofore unique stellar system - a close, massive binary with a rapidly rotating, magnetized component - that will be a rich target for further study.