This article presents a limit theorem for the gaps Ĝi:n := X _n-i+1:n - X _n-i:n between order statistics X _1:n≤ · · · ≤ X _n:n of a sample of size n from a random discrete distribution on the positive integers (P ₁,P ₂, . .) governed by a residual allocation model (also called a Bernoulli sieve) P _j := H _j ^j-1 _i=1 (1 - Hi ) for a sequence of independent random hazard variables Hi which are identically distributed according to some distribution of H ϵ (0, 1) such that -log(1 - H) has a non-lattice distribution with finite mean μlog. As n→∞the finite dimensional distributions of the gaps Ĝ+ _i:n converge to those of limiting gaps Gi which are the numbers of points in a stationary renewal process with i.i.d. spacings -log(1-H _j ) between times T _i-1 and T _i of births in a Yule process, that is T _i:=Σ ⁱ _{k=1 ek}/k for a sequence of i.i.d. exponential variables ϵ _k with mean 1. A consequence is that the mean of Ĝ+ _i:n converges to the mean of G _i, which is 1/(iμ _log). This limit theorem simplifies and extends a result of Gnedin, Iksanov and Roesler for the Bernoulli sieve.