  Consider an urn containing balls labeled with integer values. Define a
discrete-time random process by drawing two balls, one at a time and with
replacement, and noting the labels. Add a new ball labeled with the sum of the
two drawn labels. This model was introduced by Siegmund and Yakir ({\it Ann.
Probab. 33(5), 2005}) for labels taking values in a finite group, in which case
the distribution defined by the urn converges to the uniform distribution on
the group. For the urn of integers, the main result of this paper is an
exponential limit law. The mean of the exponential is a random variable with
distribution depending on the starting configuration. This is a novel urn model
which combines multi-drawing and an infinite type of balls. The proof of
convergence uses the contraction method for recursive distributional equations.
