Around 0.5 K, the entropy of the spin-ice Dy$_2$Ti$_2$O$_7$ has a plateau-like feature close to Paulings residual entropy derived originally for water ice, but an unambiguous quantification towards lower temperature is prevented by ultra-slow thermal
equilibration. Based on specific heat data of (Dy$_{1-x}$Y$_x$)$_2$Ti$_2$O$_7$ we analyze the influence of non-magnetic dilution on the low-temperature entropy. With increasing x, the ultra-slow thermal equilibration rapidly vanishes, the low-temperature entropy systematically decreases and its temperature dependence strongly increases. These data suggest that a non-degenerate ground state is realized in (Dy$_{1-x}$Y$_x$)$_2$Ti$_2$O$_7$ for intermediate dilution. This contradicts the expected zero-temperature residual entropy obtained from a generalization of Paulings theory for dilute spin ice, but is supported by Monte Carlo simulations.
