  We report the discovery of an eccentric hot Neptune and a non-transiting
outer planet around TOI-1272. We identified the eccentricity of the inner
planet, with an orbital period of 3.3 d and $R_{\rm p,b} = 4.1 \pm 0.2$
$R_\oplus$, based on a mismatch between the observed transit duration and the
expected duration for a circular orbit. Using ground-based radial velocity
measurements from the HIRES instrument at the Keck Observatory, we measured the
mass of TOI-1272b to be $M_{\rm p,b} = 25 \pm 2$ $M_\oplus$. We also confirmed
a high eccentricity of $e_b = 0.34 \pm 0.06$, placing TOI-1272b among the most
eccentric well-characterized sub-Jovians. We used these RV measurements to also
identify a non-transiting outer companion on an 8.7-d orbit with a similar mass
of $M_{\rm p,c}$ sin$i= 27 \pm 3$ $M_\oplus$ and $e_c \lesssim 0.35$.
Dynamically stable planet-planet interactions have likely allowed TOI-1272b to
avoid tidal eccentricity decay despite the short circularization timescale
expected for a close-in eccentric Neptune. TOI-1272b also maintains an envelope
mass fraction of $f_{\rm env} \approx 11\%$ despite its high equilibrium
temperature, implying that it may currently be undergoing photoevaporation.
This planet joins a small population of short-period Neptune-like planets
within the "Hot Neptune Desert" with a poorly understood formation pathway.
