We describe planetesimal accretion calculations in the Kuiper Belt. Our evolution code simulates planetesimal growth in a single annulus and includes velocity evolution but not fragmentation. Test results match analytic solutions and duplicate previous simulations at 1 AU. In the Kuiper Belt, simulations without velocity evolution produce a single runaway body with a radius r_i >~ 1000 km on a timscale tau_r~M^-1_0e^x_0, where M_0 is the initial mass in the annulus, e_0 is the initial eccentricity of the planetesimals, and x ~ 1-2. Runaway growth occurs in 100 Myr for M_0 ~ 10M_E and e_0 ~ 10^-3 in a 6 AU annulus centered at 35 AU. This mass is close to the amount of dusty material expected in a minimum-mass solar nebula extrapolated into the Kuiper Belt. Simulations with velocity evolution produce runaway growth on a wide range of timescales. Dynamical friction and viscous stirring increase particle velocities in models with large (8 km radius) initial bodies. This velocity increase delays runaway growth by a factor of 2 compared with models without velocity evolution. In contrast, collisional damping dominates over dynamical friction and viscous stirring in models with small (80-800 m) initial bodies. Collisional damping decreases the timescale to runaway growth by factors of 4-10 relative to constant-velocity calculations. Simulations with minimum-mass solar nebulae, M_0 ~ 10M_E, and small eccentricities, e ~ 10^-3, reach runaway growth on timescales of 20-40 Myr with 80 m initial bodies, 50-100 Myr with 800 m bodies, and 75-250 Myr for 8 km initial bodies. These growth times vary linearly with the mass of the annulus, tau_r~M^-1_0, but are less sensitive to the initial eccentricity than constant-velocity models. In both sets of models, the timescales to produce 1000+ km objects are comparable to estimated formation timescales for Neptune. Thus, Pluto-sized objects can form in the outer solar system in parallel with the condensation of the outermost large planets.