Plaskett's "star" appears to be one of a small number of short-period binary systems known to contain a hot, massive, magnetic star. Building on the 2013 discovery investigation, we combine an extensive spectropolarimetric (Stokes $V$) dataset 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 timeseries 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 $1.21551^{+0.00028}_{-0.00034}$ d. Modeling 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, but with an important quadrupolar component, and weak higher order components. The dipolar component has an obliquity near 90 deg and a polar strength of about 850 G, while the average field strength over the entire surface is 520 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.