A method is presented to construct a fully developed elongational flow at a nearly constant extension rate in a converging microchannel. For a Giesekus-Leonov fluid, we show that under appropriate conditions, the first normal stress difference in a fluid element flowing along the channel centerline reaches its steady-state value early in the converging region, so that the time-averaged normal stress difference is approximately equal to the spatially averaged normal stress along the converging section. We demonstrate that the averaged normal stress in the converging region (with contraction ratios of 4.9 or 10.343) maintains greater than 90% of the steady value up to an extension rate of 100 s(-)1. At higher extension rates, the averaged normal stress becomes significantly smaller than the steady value. A differential pressure elongational rheometer is proposed where the pressure difference between the converging channel and a reference channel can be monitored. The reference channel is a straight channel geometry in which the viscous contribution to the pressure drop equals that in the converging channel. (C) 2017 The Society of Rheology.