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The classical Maxwell--Born--Infeld field equations coupled with aHamilton--Jacobi law of point charge motion are partially quantized by couplingthe Hamilton-Jacobi phase function with an amplitude function, which combineswith the phase function into a single complex wave function satisfying arelativistic Klein--Gordon equation self-consistently coupled to the evolutionequations for the electromagnetic fields with generic point source (explicitlyworked out for one particle; options for many particles briefly discussed).Radiation-free stationary states exist. The hydrogen spectrum with infinitelymassive nucleus is discussed in some detail and upper estimates for Born's`aether constant' obtained. In the nonrelativistic limit the model reduces tothe de-Broglie--Bohm formulation of quantum mechanics.Comment: Corrections at galley stage incorporated (mostly minor corrections,  except for a blunder in the estimate of the error term U to the Coulomb  interaction) 38p; to appear in JSP vol. 116, issue dedicated to Elliott H.  Lieb on his 70th birthday. Part I is math-ph/030607

Electromagnetic Field Theory Without Divergence Problems 2. A Least Invasively Quantized Theory