We present constraints on the dark energy equation-of-state parameter, w = P/(pc super(2)), using 60 SNe Ia from the ESSENCE supernova survey. We derive a set of constraints on the nature of the dark energy assuming a flat universe. By including constraints on ( Omega sub(M), w) from baryon acoustic oscillations, we obtain a value for a static equation-of-state parameter w = -1.05 super(+) sub(-) super(0) sub(0) super(.) sub(.) super(1) sub(1) super(3) sub(2) (stat 1 sigma ) plus or minus 0.13 (sys) and Omega sub(M) = 0.274 super(+) sub(-) super(0) sub(0) super(.) sub(.) super(0) sub(0) super(3) sub(2) super(3) sub(0) (stat 1 sigma ) with a best-fit X super(2)/dof of 0.96. These results are consistent with those reported by the Supernova Legacy Survey from the first year of a similar program measuring supernova distances and redshifts. We evaluate sources of systematic error that afflict supernova observations and present Monte Carlo simulations that explore these effects. Currently, the largest systematic with the potential to affect our measurements is the treatment of extinction due to dust in the supernova host galaxies. Combining our set of ESSENCE SNe Ia with the first-results Supernova Legacy Survey SNe Ia, we obtain a joint constraint of w = -1.07 super(+) sub(-) super(0) sub(0) super(.) sub(.) super(0) sub(0) super(9) sub(9) (stat 1 sigma ) plus or minus 0.13 (sys), Omega sub(M) = 0.267 super(+) sub(-) super(0) sub(0) super(.) sub(.) super(0) sub(0) super(2) sub(1) super(8) sub(8) (stat 1 sigma ) with a best-fit X super(2)/dof of 0.91. The current global SN Ia data alone rule out empty ( Omega sub(M) = 0), matter-only Omega sub(M) = 0.3, and Omega sub(M) = 1 universes at >4.5 sigma . The current SN Ia data are fully consistent with a cosmological constant.