Summary. An exact conditional test for an M-way log-linear interaction in a fully observed 2M contingency table is formulated. Prom this is derived a procedure for interval estimation of the total count N in a 2 M contingency table, one of whose entries is unobserved. This procedure has an immediate application to interval estimation of the size of a closed population from incomplete, overlapping lists of records, as in capture-recapture analysis of epidemiological data. Data on the prevalence of spina bifida in live births in upstate New York in 1969–1974 illustrate this application.