At that time the French Academy of Sciences was organizing a major scientific expedition to Quito, in present-day Ecuador, in order to measure the length of a degree of meridian arc (i.e., latitude) at the equator.  This was part of an effort to determine in the precise figure of the Earth in order to settle the scientific debate between the defenders of René Descartes's physics and those who advocated the newer Newtonian mechanics.  In 1735, Ulloa and another young naval officer, Jorge Juan, were appointed by the Spanish Crown to accompany the French Geodesic Mission to Quito.  This was a sensitive assignment, both politically and scientifically.
The early work of the French Geodesic Mission, led by Charles Marie de La Condamine, was delayed and hindered by lack of cooperation from the local Spanish authorities.  Indeed, in 1737 the personal dispute between Ulloa and the president of the Real Audiencia de Quito, Joseph de Araujo y Río, reached such a pitch that Araujo ordered the arrest of Ulloa and Juan, announcing his intention to have them killed. Juan and Ulloa took refuge in a church and Ulloa then escaped through the cordon of Araujos men, reaching Lima and obtaining the protection of the Viceroy of Peru, the Marquis of Villagarcía. When war between Spain and Great Britain broke out in 1739 (see War of Jenkins Ear), Juan and Ulloa, as naval officers, actively participated in the defense of Peru.
Ulloa traveled throughout the territories of the Viceroyalty of Peru from 1736 to 1744, making many astronomic, natural, and social observations.  In one of his reports he described, for the first time in the European scientific literature, some of the properties of a metal that he called platina ("little silver") and which he encountered during his inspection the gold panning operations in the Chocó region of what is now Colombia.  This metal would later be identified as a new chemical element, now known as platinum. Ulloa is therefore often credited as the discoverer of platinum. 
The final results of French Geodesic Mission, published by La Condamine in 1745, combined with the measurements of meridian arc near the Arctic Circle that had been published in 1738 by Pierre Louis Maupertuis following the French Geodesic Mission to Lapland, decisively vindicated the predictions first made by Isaac Newton in Book III of his Principia Mathematica of 1687.  These results greatly contributed to the triumph of Newtonianism over Cartesianism among Continental European savants.