The MACHO Project is a search for dark matter in the form of massive compact halo objects (MACHOs). Photometric monitoring of millions of stars in the Large Magellanic Cloud (LMC), Small Magellanic Cloud (SMC), and Galactic bulge is used to search for gravitational microlensing events caused by these otherwise invisible objects. Analysis of the first 2.1 yr of photometry of 8.5 million stars in the LMC reveals eight candidate microlensing events. This is substantially more than the number expected (~1.1) from lensing by known stellar populations. The timescales ( t ) of the events range from 34 to 145 days. We estimate the total microlensing optical depth toward the LMC from events with 2 < ##IMG## [http://ej.iop.org/icons/Entities/hatt.gif] hat t < 200 days to be τ 200 2 =2.9 + 1.4 −0.9 ×10 −7 based upon our eight event sample. This exceeds the optical depth, τ backgnd = 0.5 × 10 -7 , expected from known stars, and the difference is to be compared with the optical depth predicted for a "standard" halo composed entirely of MACHOs: τ halo = 4.7 × 10 -7 . To compare with Galactic halo models, we perform likelihood analyses on the full eight-event sample and a six-event subsample (which allows for two events to be caused by a nonhalo "background"). This gives a fairly model-independent estimate of the halo mass in MACHOs within 50 kpc of 2.0 + 1.2 −0.7 ×10 11 M ☉ , which is about half of the "standard halo" value. We also find a most probable MACHO mass of 0.5 + 0.3 −0.2 M ☉ , although this value is strongly model dependent. In addition, the absence of short duration events places stringent upper limits on the contribution of low-mass MACHOs: objects from 10 -4 M ☉ to 0.03 M ☉ contribute ##IMG## [http://ej.iop.org/icons/Entities/lesssim.gif] lesssim 20% of the "standard" dark halo.