  We investigate global persistence properties for the non-equilibrium critical
dynamics of the randomly diluted Ising model. The disorder averaged persistence
probability $\bar{{P}_c}(t)$ of the global magnetization is found to decay
algebraically with an exponent $\theta_c$ that we compute analytically in a
dimensional expansion in $d=4-\epsilon$. Corrections to Markov process are
found to occur already at one loop order and $\theta_c$ is thus a novel
exponent characterizing this disordered critical point. Our result is
thoroughly compared with Monte Carlo simulations in $d=3$, which also include a
measurement of the initial slip exponent. Taking carefully into account
corrections to scaling, $\theta_c$ is found to be a universal exponent,
independent of the dilution factor $p$ along the critical line at $T_c(p)$, and
in good agreement with our one loop calculation.
