An ultrarelativistic 28.5 GeV, 700-microm-long positron bunch is focused near the entrance of a 1.4-m-long plasma with a density n(e) between approximately equal to 10(13) and approximately equal to 5 x 10(14) cm(-3). Partial neutralization of the bunch space charge by the mobile plasma electrons results in a reduction in transverse size by a factor of approximately equal to 3 in the high emittance plane of the beam approximately equal to 1 m downstream from the plasma exit. As n(e) increases, the formation of a beam halo containing approximately 40% of the total charge is observed, indicating that the plasma focusing force is nonlinear. Numerical simulations confirm these observations. The bunch with an incoming transverse size ratio of approximately 3 and emittance ratio of approximately 5 suffers emittance growth and exits the plasma with approximately equal sizes and emittances.