  Millimeter observations of disks around young stars reveal substructures
indicative of gas pressure traps that may aid grain growth and planet
formation. We present Submillimeter Array observations of HD 34700- two Herbig
Ae stars in a close binary system (Aa/Ab, $\sim$0.25 AU), surrounded by a disk
presenting a large cavity and spiral arms seen in scattered light, and two
distant, lower mass companions. These observations include 1.3 mm continuum
emission and the $^{12}$CO 2-1 line at $\sim0.5$" (178 AU) resolution. They
resolve a prominent azimuthal asymmetry in the continuum, and Keplerian
rotation of a circumbinary disk in the $^{12}$CO line. The asymmetry is located
at a radius of $155^{+11}_{-7}$ AU, consistent with the edge of the scattered
light cavity, being resolved in both radius ($72 ^{+14}_{-15}$ AU) and azimuth
(FWHM = $64 ^{\circ +8}_{-7}$). The strong asymmetry in millimeter continuum
emission could be evidence for a dust trap, together with the more symmetric
morphology of $^{12}$CO emission and small grains. We hypothesize an unseen
circumbinary companion, responsible for the cavity in scattered light and
creating a vortex at the cavity edge that manifests in dust trapping. The disk
mass has limitations imposed by the detection of $^{12}$CO and non-detection of
$^{13}$CO. We discuss its consequences for the potential past gravitational
instability of this system, likely accounting for the rapid formation of a
circumbinary companion. We also report the discovery of resolved continuum
emission associated with HD 34700B (projected separation $\sim1850$AU), which
we explain through a circumstellar disk.
