For zero-truncated count data, as they typically arise in capture-recapture modelling, we consider modelling under one-inflation. This is motivated by several data sets including police data on drink-driving in Britain which shows high one-inflation.   We show that inference for an arbitrary count density with one-inflation  can be equivalently based upon the associated zero-one truncated count density. This simplifies inference considerably including maximum likelihood estimation and likelihood ratio testing.  Illustrations are provided for several data sets inclduing the drink-driving example.