William Mossop was born in St. Mary's parish in Dublin to a Roman Catholic father named Browne.

In 1783, he executed a medal bearing a portrait of Dr. Henry Quin, which was presented to the doctor by Robert Watson Wade, First Clerk of the Irish Treasury, as a token of gratitude for his recovery from a severe illness.

The medal bears on the obverse a portrait of Lord Charlemont in the uniform of the Irish Volunteers, and on the reverse Hibernia seated on a pile of books, surrounded by the emblems of Astronomy, Chemistry, Poetry and Antiquities.

His gratitude was expressed by an inscription on the reverse of the medal to Quin, which he had executed in 1783: "Sacred to the Man who, after finding out the Author in obscurity, led him into the profession of this polite art and became his patron, his friend and liberal benefactor.

"[2] Mossop struck a number of medals for notable people in Dublin, the finest being for James Caulfeild, 1st Earl of Charlemont.