We present calculations of the early stages of the formation of Jupiter via core nucleated accretion and gas capture. The core begins as a seed body of about 350 kilometers in radius and orbits in a swarm of planetesimals whose initial radii range from 15 meters to 50 kilometers. The evolution of the swarm accounts for growth and fragmentation, viscous and gravitational stirring, and for drag-assisted migration and velocity damping. During this evolution, less than 9% of the mass is in planetesimals smaller than 1 kilometer in radius; < ~25% is in planetesimals with radii between 1 and 10 kilometers; and < ~7% is in bodies with radii larger than 100 kilometers. Gas capture by the core substantially enhances the size-dependent cross-section of the planet for accretion of planetesimals. The calculation of dust opacity in the planet's envelope accounts for coagulation and sedimentation of dust particles released as planetesimals are ablated. The calculation is carried out at an orbital semi-major axis of 5.2 AU and the initial solids' surface density is 10 g/cm^2 at that distance. The results give a core mass of nearly 7.3 Earth masses (Mearth) and an envelope mass of approximately 0.15 Mearth after about 4e5 years, at which point the envelope growth rate surpasses that of the core. The same calculation without the envelope yields a core of only about 4.4 Mearth.