  The transiting planet HD80606b undergoes a 1000-fold increase in insolation
during its 111-day orbit due to it being highly eccentric (e=0.93). The
planet's effective temperature increases from 400K to over 1400K in a few hours
as it makes a rapid passage to within 0.03AU of its host star during periapsis.
Spectroscopic observations during the eclipse (which is conveniently oriented a
few hours before periapsis) of HD80606b with the James Webb Space Telescope
(JWST) are poised to exploit this highly variable environment to study a wide
variety of atmospheric properties, including composition, chemical and
dynamical timescales, and large scale atmospheric motions. Critical to planning
and interpreting these observations is an accurate knowledge of the planet's
orbit. We report on observations of two full-transit events: 7 February 2020 as
observed by the TESS spacecraft and 7--8 December 2021 as observed with a
worldwide network of small telescopes. We also report new radial velocity
observations which when analyzed with a coupled model to the transits greatly
improve the planet's orbital ephemeris. Our new orbit solution reduces the
uncertainty in the transit and eclipse timing of the JWST era from tens of
minutes to a few minutes. When combined with the planned JWST observations,
this new precision may be adequate to look for non-Keplerian effects in the
orbit of HD80606b.
