Birds have many physiological characteristics that are convergent with mammals. In the light of recent evidence that mammals can maintain a constant insensible evaporative water loss (EWL) over a range of perturbing environmental conditions, we hypothesized that birds might also regulate insensible EWL, reflecting this convergence. We found that budgerigars (Melopsittacus undulatus) maintain EWL constant over a range of relative humidities at three ambient temperatures. EWL, expressed as a function of water vapour pressure deficit, differed from a physical model where the water vapour pressure deficit between the animal and the ambient air is the driver of evaporation, indicating physiological control of EWL. Regulating EWL avoids thermoregulatory impacts of varied evaporative heat loss; changes in relative humidity had no effect on body temperature, metabolic rate or thermal conductance. Our findings that a small bird can regulate EWL are evidence that this is a common feature of convergently endothermic birds and mammals, and may therefore be a fundamental characteristic of endothermy. © 2017 The Author(s).