  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.
