  We discuss exclusive electromagnetic production of two neutral $\rho$ mesons
in coherent photon-photon processes in ultrarelativistic heavy-ion collisions.
The cross section is calculated in the equivalent photon approximation (EPA).
Both uncertainties of the flux factors and photon-photon cross sections are
discussed in details. We show that inclusion of precise charge densities in
nuclei is essential for realistic evaluations of the nuclear photon-photon
cross sections. We find that the cross section, especially with realistic flux
factors, is sensitive to low energy in the subsystem $\gamma \gamma \to \rho^0
\rho^0$. The experimental data for the $\gamma \gamma \to \rho^0 \rho^0$ cross
section extracted from $e^+ e^-$ collisions are parametrized and used to
estimate the nucleus-nucleus cross section. In addition, we include
vector-dominance-model(VDM)--Regge contribution which becomes important at
large photon-photon energy. Large nuclear cross sections are obtained. We
discuss a possibility of focusing on the large-energy component. We find that
both $\rho^0$ mesons are produced predominantly at midrapidities and could be
measured by the STAR collaboration at RHIC.
