  It is shown that the pi^0 transition form factor F(Q_1^2,Q_2^2) differs
substantially from its one-real-photon limit F(Q_1^2,0) even for rather small
values of Q_2^2 (approx 0.1 GeV^2), which cannot be excluded in experiments
with one "untagged" electron. It indicates that the comparison of data with
theoretical calculations, which usually assume Q_2^2=0, may be untrustworthy.
Our phenomenological model of the pi^0 transition form factor is based on the
vector-meson-dominance hypothesis and all its parameters are fixed by using the
experimental data on the decays of vector mesons. The model soundness is
checked in the two-real-photon limit, where it provides a good parameter-free
description of the pi^0 -> 2 gamma decay rate, and in the pi^0 Dalitz decay.
The dependence of F(Q_1^2,Q_2^2) on Q_1^2 at several fixed values of Q_2^2 is
presented and the comparison with existing data performed.
