This Eyewitness to History post describes the event. The monarch was attended the night before and the day of his execution by an Anglo-Irish priest, the Abbe Henry Essex Edgeworth, who heard the king's last Confession, celebrated Mass for him, and supported him throughout the ordeal--he is the eyewitness cited in the link above. According to 1878's A Compendium of Irish Biography found on this site, he was a:
cousin of Richard Lovell Edgeworth, was born at Edgeworthstown in 1745. His father, Essex Edgeworth, "who took the name of "de Firmont" from a neighbouring hill (Fairy Mount), became a Catholic and emigrated to France when Henry was but six years of age. The lad was educated for the priesthood at the Sorbonne, and after ordination became distinguished among the Parisian clergy for his talents and piety. In 1789 he was appointed confessor to Madame Elizabeth, and was justly esteemed the friend and adviser of the royal family. When Louis XVI. was condemned to the guillotine, he sent for the Abbe Edgeworth, then in concealment at Choisy, who immediately repaired to his master. The Abbe attended the unfortunate King to the scaffold, 21st January 1793, and has left a minute account of the execution. He makes no mention of the exclamation usually attributed to him as the knife fell — "Son of St. Louis, ascend to heaven!" After encountering many dangers, he escaped to England in 1796, where he is stated to have declined a pension offered him by Pitt. He afterwards joined Louis XVIII. at Blankenburg, and accompanied him to Mittau. He was from time to time intrusted with several important missions for the Bourbons. He fell a victim to a virulent fever, caught in his ministrations amongst French prisoners of war at Mittau, and died 22nd May 1807, aged about 62. In his last moments he was attended by the Princess, daughter of Louis XVI.; the exiled French royal family went into mourning, and Louis XVIII. composed his epitaph.
The town of Mitau was bright with snow in the sunshine of a May morning, and cold winds whipped around the little palace which His Imperial Majesty the Tsar of All the Russias had generously loaned to the impoverished, exiled Bourbons. In a small, sparsely furnished room of the palace, an aged priest lay dying. In a chair beside his bed sat a young woman, not quite thirty, in a maroon, high-waisted wool dress, with a white linen apron. Under a close fitting white cap, her amber-colored hair, in a grecian knot, framed a strong, solemn face with piercing blue-gray eyes. Dipping a cloth into a basin of water, she sponged the forehead of the sick man, whose chest shuddered and heaved. At first glance, no one would guess that she was Madame Royale, daughter of Louis XVI and Marie-Antoinette, their Mousseline la sérieuse, now the Duchesse d’Angoulême. With closer examination, no one with her dignified, albeit rather stiff bearing could be anything but a princess. She radiated a cold majesty to those who did not know her, but in her eyes burned the fires of deep emotions; her frigid manner was from sadness, not apathy or scorn.
The Catholic Encyclopedia notes that the Edgeworth family moved to Ireland during the reign of Elizabeth I and the Abbe's father became a Catholic during the reign of King George II. Edgeworth was offered an Irish diocese after his ordination but noted that England was a distant second language to him since he was raised in France from childhood. When Le Clerc, the Archbishop of Paris, fled in 1792, he appointed Edgeworth as the Vicar General of the diocese, and the Holy See never recognized Jean-Baptiste-Joseph Gobel, who had taken the oath of the Civil Constitution of the Clergy (who ended up on the guillotine just like Louis XVI--on April 13, 1794).