The cosmological star formation rate in the combined Chandra Deep Fields North and South is derived from our X-ray luminosity function for galaxies in these deep fields. Mild evolution is seen up to redshift order unity with star formation rate ~ (1 + z)2.7. This is the first directly observed normal star-forming galaxy X-ray luminosity function (XLF) at cosmologically interesting redshifts (z > 0). This provides the most direct measure yet of the X-ray-derived cosmic star formation history of the universe. We make use of Bayesian statistical methods to classify the galaxies and the two types of active galactic nuclei (AGNs), finding the most useful discriminators to be the X-ray luminosity, X-ray hardness ratio, and X-ray to optical flux ratio. There is some residual AGN contamination in the sample at the bright end of the luminosity function. Incompleteness slightly flattens the XLF at the faint end of the luminosity function. The XLF has a lognormal distribution and agrees well with the radio and infrared luminosity functions. However, the XLF does not agree with the Schechter luminosity function for the Hα LF, indicating that, as discussed in the text, additional and different physical processes may be involved in the establishment of the lognormal form of the XLF. The agreement of our star formation history points with the other star formation determinations in different wavebands (IR, radio, Hα) gives an interesting constraint on the initial mass function (IMF). The X-ray emission in the Chandra band is most likely due to binary stars, although X-ray emission from nonstellar sources (e.g., intermediate-mass black holes and/or low-luminosity AGNs) remains a possibility. Under the assumption that it is due to binary stars, the overall consistency and correlations between single-star effects and binary-star effects indicate that not only is the one-parameter IMF (M) constant but also the bivariate IMF(M1, M2) must be constant, at least at the high-mass end. Another way to put this, quite simply, is that X-ray observations may be measuring directly the binary-star formation history of the universe. X-ray studies will continue to be useful for probing the star formation history of the universe by avoiding problems of obscuration. Star formation may therefore be measured in more detail by deep surveys with future X-ray missions.