We have obtained HST and ground-based observations of a sample of 20 O-type stars in the LMC and SMC, including six of the hottest massive stars known (subtypes O2-3) in the R136 cluster. In general, these data include (a) the HST UV spectra in order to measure the terminal velocities of the stellar winds, (b) high signal-to-noise, blue-optical data where the primary temperature- and gravity-sensitive photospheric lines are found, and (c) nebular-free H-alpha profiles, which provide the mass-loss rates. The line-blanketed non-LTE atmosphere code FASTWIND was then used to determine the physical parameters of this sample of stars. We find good agreement between the synthetic line profiles for the hydrogen, He I, and He II lines in the majority of the stars we analyzed; the three exceptions show evidence of being incipiently resolved spectroscopic binaries or otherwise spectral composites. One such system is apparently an O3 V+O3 V eclipsing binary, and a follow-up radial velocity study is planned to obtain Keplerian masses. Although we did not use them to constrain the fits, good agreement is also found for the He I $\lambda 3187$ and He II $\lambda 3203$ lines in the near-UV, which we plan to exploit in future studies. Our effective temperatures are compared to those recently obtained by Repolust, Puls & Herrero for a sample of Galactic stars using the same techniques. We find that the Magellanic Cloud sample is 3,000-4,000$^\circ$K hotter than their Galactic counterparts for the early through mid-O's. These higher temperatures are the consequence of a decreased importance of wind emission, wind blanketing, and metal-line blanketing at lower metallicities. Comment: Accepted for publication in the Astrophysical Journal. A postscript version with the figures embedded can be found at ftp://ftp.lowell.edu/pub/massey/haw.ps