  We report on a search for ultra-high-energy (UHE) neutrinos from gamma-ray
bursts (GRBs) in the data set collected by the Testbed station of the Askaryan
Radio Array (ARA) in 2011 and 2012. From 57 selected GRBs, we observed no
events that survive our cuts, which is consistent with 0.12 expected background
events. Using NeuCosmA as a numerical GRB reference emission model, we estimate
upper limits on the prompt UHE GRB neutrino fluence and quasi-diffuse flux from
$10^{7}$ to $10^{10}$ GeV. This is the first limit on the prompt UHE GRB
neutrino quasi-diffuse flux above $10^{7}$ GeV.
