On 2015 March 23, VERITAS responded to a $Swift$-BAT detection of a gamma-ray burst, with observations beginning 270 seconds after the onset of BAT emission, and only 135 seconds after the main BAT emission peak. No statistically significant signal is detected above 140 GeV. The VERITAS upper limit on the fluence in a 40 minute integration corresponds to about 1% of the prompt fluence. Our limit is particularly significant since the very-high-energy (VHE) observation started only $sim$2 minutes after the prompt emission peaked, and $Fermi$-LAT observations of numerous other bursts have revealed that the high-energy emission is typically delayed relative to the prompt radiation and lasts significantly longer. Also, the proximity of GRB~150323A ($z=0.593$) limits the attenuation by the extragalactic background light to $sim 50$ % at 100-200 GeV. We conclude that GRB 150323A had an intrinsically very weak high-energy afterglow, or that the GeV spectrum had a turnover below $sim100$ GeV. If the GRB exploded into the stellar wind of a massive progenitor, the VHE non-detection constrains the wind density parameter to be $Agtrsim 3times 10^{11}$ g cm$^{-1}$, consistent with a standard Wolf-Rayet progenitor. Alternatively, the VHE emission from the blast wave would be weak in a very tenuous medium such as the ISM, which therefore cannot be ruled out as the environment of GRB 150323A.
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