Electroproduction of relativistic positronium may be of importance for obtaining beams of polarized or unpolarized positronium. We give cross sections for singlet and triplet positronium production in high-energy collisions of electrons on atoms, correct to all orders in the atomic number Z, including a complete description of atomic screening, using the Thomas-Fermi-Molière atomic model. The results are analogous to the photoproduction cross section results recently published [Phys. Rev. A 60, 1883 (1999)]. However, since in electroproduction the positronum is produced as a spectrum with spectral distributions varying considerably according to the spin of the positronium, singlet or triplet, the resulting cross sections are strongly dependent on screening, more so than in photoproduction. We show that with initially longitudinaly polarized electrons, polarization may be transferred to the positronium, so that beams of sizably polarized positronium may be obtained.