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.