Count data arise in various organizational settings. When the release of such data is sensitive, organizations need information-disclosure policies that protect data confidentiality while still providing data access. In contrast to extant disclosure policies, we describe a new policy for count tables that is based on disclosing only the sufficient statistics of a flexible discrete distribution. This distribution, the COM-Poisson, well approximates Poisson counts but also under- and over-dispersed counts. The sufficient statistics mask the exact cell counts and often also the table size. Under the scenario of a data holding agency and a data snooper, we show that this policy has low disclosure risk with no loss of data utility: Usually, many count tables correspond to the disclosed sufficient statistics. Furthermore, these count tables are equally likely to be the undisclosed table. Finding these solutions requires solving a system of linear equations, which are underdetermined for tables with more than three cells, and can be computationally prohibitive for even small tables. We also consider cell-specific interval bounds, a commonly used disclosure limitation policy, and compare them to our policy. We describe several types of snooper knowledge, their integration with the disclosed statistics, and implications. Applying this policy to three real data sets, we illustrate the low associated disclosure risk.