  We measure the dependence of $\bar B^0 \to D^{*+}e^-\bar{\nu}_e$ on the decay
angles and momentum transfer. The data sample consists of $\sim 86 \times 10^6$
$B\bar B$-pairs accumulated on the $\Upsilon(4{\rm S})$ resonance by the BABAR
detector at the asymmetric $e^+ e^-$ collider PEP-II. We specify the three form
factors by two ratios $R_1$ and $R_2$, and by a single parameter $\rho^2$
characterizing the polynomial representing $h_{A_1}$, the function which
describes the momentum-transfer dependence of the form factor $A_1$. We
determine $R_1$, $R_2$, and $\rho^2$ using an unbinned maximum likelihood fit
to the full decay distribution. The results are $R_1=1.396\pm 0.060\pm 0.035\pm
0.027$, $R_2=0.885\pm 0.040\pm 0.022\pm 0.013$, and $\rho^2=1.145\pm 0.059\pm
0.030\pm 0.035$. 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 $|V_{cb}|$.
