Nuclear E0-transitions above the pair production threshold of co--2rn c2,-, 1 MeV are considered. The ration r/of the electron-positron pair creation probability to the K-shell conversion probability is calculated. For 92U and co= 1,450keV a typical value of t/ =2,10 .3 is obtained. Good agreement with experimental data is found. Our numerical results for finite size nuclei are compared with analytical expressions utilizing relativistic electron wavefunctions for point nuclei. The energy distributions of emitted positrons are presented for various nuclear charge values and transition energies. The possibility of monoenergetic positron emission is discussed. Of particular interest is the relation of the theoretical results to the recently observed structure in the positron spectra, which have been measured in high energy collisions of very heavy ions.