The evaporation of falling diethyl ether droplets is measured by following droplets along their trajectories. Measurements are performed at ambient temperature and pressure by using in-line digital holography. The holograms of droplets are recorded with a single high-speed camera and reconstructed with an ''inverse problems'' approach algorithm previously tested (Chareyron et al. New J Phys 14:43039, 2012). Once evaporation starts, the interfaces of the droplets are surrounded by air/vapor mixtures with refractive index gradients that modify the holograms. The central part of the droplets holograms is unusually bright compared to what is expected and observed for non-evaporating droplets. The reconstruction process is accordingly adapted to measure the droplets diameter along their trajectory. The diethyl ether being volatile, the droplets are found to evaporate in a very short time: of the order of 70 ms for a 50-60 lm diameter at an ambient temperature of 25 C. After this time, the diethyl ether has fully evaporated and droplets diameter reaches a plateau. The remaining droplets are then only composed of water, originating from the cooling and condensation of the humid air at the droplet surface. This assertion is supported by two pieces of evidence: (i) by estimating the evolution of droplets refractive index from light scattering measurements at rainbow angle and (ii) by comparing the evaporation rate and droplets velocities obtained by digital holography with those calculated with a simple model of evaporation/condensation. The overall results show that the in-line digital holography with ''inverse problems''approach is an accurate technique for studying fast evaporation from a Lagrangian point of view.