We have measured the nuclear transparency of the incoherent diffractive $A(e,e'\rho^0)$ process in $^{12}$C and $^{56}$Fe targets relative to $^2$H using a 5 GeV electron beam. The nuclear transparency, the ratio of the produced $\rho^0$'s on a nucleus relative to deuterium, which is sensitive to $\rho A$ interaction, was studied as function of the coherence length ($l_c$), a lifetime of the hadronic fluctuation of the virtual photon, and the four-momentum transfer squared ($Q^2$). While the transparency for both $^{12}$C and $^{56}$Fe showed no $l_c$ dependence, a significant $Q^2$ dependence was measured, which is consistent with calculations that included the color transparency effects.