We measure the dependence of B̄0→D*+e-ν̄e on the decay angles and momentum transfer. The data sample consists of ∼86×106 BB̄-pairs accumulated on the Υ(4S) resonance by the BABAR detector at the asymmetric e+e- collider PEP-II. We specify the three form factors by two ratios R1 and R2, and by a single parameter ρ2 characterizing the polynomial representing hA1, the function which describes the momentum-transfer dependence of the form factor A1. We determine R1, R2, and ρ2 using an unbinned maximum likelihood fit to the full decay distribution. The results are R1=1.396±0.060±0.035±0.027, R2=0.885±0.040±0.022±0.013, and ρ2=1.145±0. 059±0.030±0.035. The stated uncertainties are the statistical from the data, statistical from the size of the Monte Carlo sample and the systematic uncertainty, respectively. In addition, based on this measurement, we give an updated value for the CKM matrix element |Vcb|. © 2006 The American Physical Society.