The laminarity of high-current multi-MeV proton beams produced by irradiating thin metallic foils with ultraintense lasers has been measured. For proton energies >10 MeV, the transverse and longitudinal emittance are, respectively, <0.004 mm mrad and <10(-4) eV s, i.e., at least 100-fold and may be as much as 10(4)-fold better than conventional accelerator beams. The fast acceleration being electrostatic from an initially cold surface, only collisions with the accelerating fast electrons appear to limit the beam laminarity. The ion beam source size is measured to be <15 microm (FWHM) for proton energies >10 MeV.