We present constraints on the dark energy equation-of-state parameter, w=P/(rho c^2), using 60 Type Ia supernovae (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_M, w) from baryon acoustic oscillations, we obtain a value for a static equation-of-state parameter w=-1.05^{+0.13}_{-0.12} (stat; 1 sigma) +- 0.11 (sys) and Omega_M=0.274^{+0.033}_{-0.020} (stat; 1 sigma) with a best-fit chi^2/DoF of 0.96. These results are consistent with those reported by the SuperNova Legacy Survey in 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 currently 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 SuperNova Legacy Survey SNe Ia, we obtain a joint constraint of w=-1.07^{+0.09}_{-0.09} (stat; 1 sigma) +- 0.12 (sys), Omega_M=0.267^{+0.028}_{-0.018} (stat; 1 sigma) with a best-fit chi^2/DoF of 0.91. The current SN Ia data are fully consistent with a cosmological constant.
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