Fermilab-Tevatron. Measurement of differential cross sections for Z/GAMMA* + JET +X production in PBAR P collisions at a centre-of-mass energy of 1.96 TeV. Data are presented in bins of the azimuthal angle between the Z/GAMMA* (P=3) and the leading jet (P=4), the absolute value of the rapidity difference between the Z/GAMMA* and the leading jet and the absolute value of the average rapidity of the Z/GAMMA* and the leading jet, all normalised to the inclusive Z/GAMMA* production cross section. Data are presented for two minimum values of the Z/GAMMA* transverse momentum. The data were collected during the period from April 2002 to February 2006 and have a total integrated luminsoity of 0.97 +- 0.06 fb-1. This record also contains, at the request of the authors, the absolute cross sections obtained by multiplying by the measured total inelastic cross section od 118 PB. This also introduces an 8% overall normalization error The systematic (DSYS) error in the first 6 tables is the total systematic error including both uncorrelated and correlated sources. The later tables give the percentage systematic errors from the individual different sources where the errors from sources 1-10 are correlated across all bins and all distributions. The sources of systematic error are: 0: Uncorrelated systematic error. 1: del(phi),del(y) and y re-weighting. 2: Jet y re-weighting. 3: Z0/GAMMA* re-weighting. 4: Jet PT re-weighting. 5: Data overlay on simulation. 6: Muon 1/PT resolution. 7: Muon identification efficiency. 8: Jet identification efficiency. 9: Jet energy resolution. 10: Jet energy scale.