  We investigate the discovery potential of the $a_1(1260)$ photoproduction in
the $\gamma p \to a_1(1260)^+ n$ and $\gamma p \to \pi^+\pi^+\pi^- n$ reactions
via the $\pi$-exchange mechanism. For the $\gamma p \to \pi^+\pi^+\pi^- n$
reaction, we perform a calculation for the differential and total cross
sections by including the contributions from the $a_1(1260)$ intermediate
resonance decaying into $\rho \pi$ then into $\pi^+\pi^+\pi^-$. Besides, the
non-resonance process is also considered. With a lower mass of $a_1(1260)$, we
get a fairly good description of the experimental data for the invariant
$\pi^+\pi^+\pi^-$ mass distributions. For the $\gamma p \to a_1(1260)^+ n$
reaction, with our model parameters, the total cross section is of the order of
10 $\mu$b at photon beam energy $E_{\gamma} \sim$ 2.5 GeV. It is expected that
our model calculations could be tested by future experiments.
