We introduce a HST/STIS stellar census of R136a, the central ionizing star cluster of 30 Doradus. We present low resolution far-ultraviolet STIS/MAMA spectroscopy of R136 using 17 contiguous 52x0.2 arcsec slits which together provide complete coverage of the central 0.85 parsec (3.4 arcsec). We provide spectral types of 90% of the 57 sources brighter than m_F555W = 16.0 mag within a radius of 0.5 parsec of R136a1, plus 8 additional nearby sources including R136b (O4,If/WN8). We measure wind velocities for 52 early-type stars from CIV 1548-51, including 16 O2-3 stars. For the first time we spectroscopically classify all Weigelt & Baier members of R136a, which comprise three WN5 stars (a1-a3), two O supergiants (a5-a6) and three early O dwarfs (a4, a7, a8). A complete Hertzsprung-Russell diagram for the most massive O stars in R136 is provided, from which we obtain a cluster age of 1.5+0.3_-0.7 Myr. In addition, we discuss the integrated ultraviolet spectrum of R136, and highlight the central role played by the most luminous stars in producing the prominent HeII 1640 emission line. This emission is totally dominated by very massive stars with initial masses above ~100 Msun. The presence of strong HeII 1640 emission in the integrated light of very young star clusters (e.g A1 in NGC 3125) favours an initial mass function extending well beyond a conventional upper limit of 100 Msun. We include montages of ultraviolet spectroscopy for LMC O stars in the Appendix. Future studies in this series will focus on optical STIS/CCD medium resolution observations.
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