A beam of K2 mesons was produced by passing a beam of 1.1-Bev/c negative pions through a liquid hydrogen target and accepting the neutral reaction products in the forward direction after allowing the K1 component to decay. The resultant beam was observed in a 30-in. propane bubble chamber fitted with lead and iron plates. About 200 regenerated K1 mesons were identified by their characteristic Q value and decay rate. All three types of regeneration were observed: by transmission in the plates, by nuclear diffraction, and by interaction with single nucleons. The detection of the first two types of regeneration constitutes strong evidence for the correctness of the Gell-Mann and Pais particle mixture theory. Comparison of the transmission and diffraction regeneration effect, using the method of M. L. Good, gives the K1-K2 mass difference δ. Two important corrections must be applied to Good's formula: One originates from the nuclear scattering of the transmission component, the other from the multiplicity of scatterings in a thick plate. The independence from nuclear parameters, which was an advantageous property of Good's formula, is no longer rigorously valid; but due to the sharp dependence of the transmission intensity upon the mass difference, the nuclear properties of K0 and K¯0, as derived from K+ and K- data, still allow a measurement of δ. We find that δ is 0.84+0.29-0.22 in units of ℏτ1, where τ1 is the K1 mean lifetime. With 90% confidence level, the difference is between 0.44 and 1.2 ℏτ1. The probability that the transmission peak we observe is due to a statistical fluctuation is one in a million.