I'm finding information that attributes this to Sir Humphrey Lloyd, who looked into it at the beckoning of none other than Sir William Rowan Hamilton.
From the Encyclopedia Britannica:
In applying his methods in 1832 to the study of the propagation of light in anisotropic media, in which the speed of light is dependent on the direction and polarization of the ray, Hamilton was led to a remarkable prediction: if a single ray of light is incident at certain angles on a face of a biaxial crystal (such as aragonite), then the refracted light will form a hollow cone.
That's the theoretical background. Lloyd then performed an experiment:
Hamilton’s colleague Humphrey Lloyd, professor of natural philosophy at Trinity College, sought to verify this prediction experimentally. Lloyd had difficulty obtaining a crystal of aragonite of sufficient size and purity, but eventually he was able to observe this phenomenon of conical refraction. This discovery excited considerable interest within the scientific community and established the reputations of both Hamilton and Lloyd.
This page also attributes the theory to Hamilton and the experiment to Lloyd:
Hamilton described his prediction when he presented the concluding part of his Third Supplement to an Essay on the Theory of Systems of Rays to the Royal Irish Academy on the 22nd of October 1832. He asked Humphrey Lloyd, the Professor of Natural and Experimental Philosophy at Trinity College, Dublin, to try to verify the prediction experimentally. Lloyd had difficulty in obtaining a suitable crystal, but, after obtaining a good specimen of arragonite, succeeded in observing external conical refraction on the 14th of December. He was subsequently able to observe internal conical refraction. Humphrey Lloyd gave an account of his experiments in two papers published in the London and Edinburgh Philosophical Magazine in February and March, 1833.
Finally, from here:
In two months, Humphrey Lloyd succeeded at observing the predicted light ring of conical refraction. In the early 1830s he used sunlight and a natural crystal.
And a confirmation of Hamilton's part:
Therefore Hamilton asked Humphrey Lloyd for experimental approval, and two months later Lloyd observed the predicted light ring. This discovery was startling to the scientists and was a triumph for the wave theory.