  We present results derived from the first multi-chord stellar occultation by
the trans-Neptunian object (229762) 2007 UK$_{126}$, observed on 2014 November
15. The event was observed by the Research and Education Collaborative
Occultation Network (RECON) project and International Occultation Timing
Association (IOTA) collaborators throughout the United States. Use of two
different data analysis methods obtain a satisfactory fit to seven chords,
yelding an elliptical fit to the chords with an equatorial radius of
$R=338_{-10} ^{+15}$ km and equivalent radius of $R_{eq}=319_{-7} ^{+14}$ km. A
circular fit also gives a radius of $R=324_{-23} ^{+30}$ km. Assuming that the
object is a Maclaurin spheroid with indeterminate aspect angle, and using two
published absolute magnitudes for the body, we derive possible ranges for
geometric albedo between $p_{V}=0.159_{-0.013} ^{+0.007}$ and
$p_{R}=0.189_{-0.015}^{+0.009}$, and for the body oblateness between
$\epsilon=0.105_{-0.040} ^{+0.050}$ and $\epsilon=0.118_{-0.048} ^{+0.055}$.
For a nominal rotational period of 11.05 h, an upper limit for density of
$\rho=1740$ kg~m$^{-3}$ is estimated for the body.
