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Plaskett's 'star' appears to be one of a small number of short-period binary systems known to contain a hot, massive, magnetic star. We combine an extensive spectropolarimetric (Stokes V) data set with archival photometry and spectropolarimetry to establish the essential characteristics of the magnetic field and magnetosphere of the rapidly rotating, broad-line component of the system. We apply least-squares deconvolution (LSD) to infer the longitudinal magnetic field from each Stokes V spectrum. Using the time series of longitudinal field measurements, in combination with CoRoT photometry and equivalent width measurements of magnetospheric spectral lines, we infer the rotation period of the magnetic star to be equal to 1.21551(-0.00034)(+0.00028) d. Modelling the Stokes V LSD profiles with Zeeman-Doppler Imaging, we produce the first reliable magnetic map of an O-type star. We find a magnetic field that is predominantly dipolar, with an obliquity near 90 degrees and a polar strength of about 850 G. We update the calculations of the theoretical magnetospheric parameters, and in agreement with their predictions we identify clear variability signatures of the H alpha, H beta, and He II lambda 4686 lines confirming the presence of a dense centrifugal magnetosphere surrounding the star. Finally, we report a lack of detection of radial velocity (RV) variations of the observed Stokes V profiles, suggesting that historical reports of the large RV variations of the broad-line star's spectral lines may be spurious. This discovery may motivate a fundamental revision of the historical model of the Plaskett's star as a near-equal mass O + O binary system.