We consider a standard two-source model for uniform common randomness (UCR) generation, in which Alice and Bob observe independent and identically distributed (i.i.d.) samples of a correlated finite source and where Alice is allowed to send information to Bob over an arbitrary single-user channel. We study the \(\boldsymbolε\)-UCR capacity for the proposed model, defined as the maximum common randomness rate one can achieve such that the probability that Alice and Bob do not agree on a common uniform or nearly uniform random variable does not exceed \(\boldsymbolε.\) We establish a lower and an upper bound on the \(\boldsymbolε\)-UCR capacity using the bounds on the \(\boldsymbolε\)-transmission capacity proved by Verdú and Han for arbitrary point-to-point channels.