  Around 0.5 K, the entropy of the spin-ice Dy$_2$Ti$_2$O$_7$ has a
plateau-like feature close to Pauling's 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 Pauling's theory for dilute spin ice, but is supported by Monte Carlo
simulations.
