Hot Neptunes, gaseous planets smaller than Saturn (∼ 3-8 R⊕) with orbital periods less than 10 days, are rare. Models predict this is due to high-energy stellar irradiation stripping planetary atmospheres over time, often leaving behind only rocky planetary cores. We present the discovery of a 6.2 R_Earth (0.55 R_J), 19.2 M_Earth(0.060 M_J) planet transiting a red giant star every 4.21285 days. The old age and high equilibrium temperature yet remarkably low density of this planet suggests that its gaseous envelope should have been stripped by high-energy stellar irradiation billions of years ago. The present day planet mass and radius suggest atmospheric stripping was slower than predicted. Unexpectedly low stellar activity and/or late-stage planet inflation could be responsible for the observed properties of this system.