Eddie "Hudson Hawk" Hawkins (Bruce Willis)—"Hudson Hawk" is a nickname for the bracing winds off the Hudson River—is a master burglar and safe-cracker, attempting to celebrate his first day of parole from prison with a cappuccino. Before he can get it, he is blackmailed by various entities, including his own parole officer, a minor Mafia family headed by the Mario Brothers, and the CIA into doing several dangerous art heists with his singing partner in crime, Tommy "Five-Tone" Messina (Danny Aiello).
The holders of the puppet strings turn out to be a "psychotic American corporation", Mayflower Industries, run by husband and wife Darwin (Richard E. Grant) and Minerva Mayflower (Sandra Bernhard) and their blade-slinging butler, Alfred (Donald Burton). The company, headquartered in the Esposizione Universale Roma, seeks to take over the world by reconstructing La Macchina dell'Oro, a machine purportedly invented by Leonardo da Vinci (Stefano Molinari) that converts lead into gold. A special assembly of crystals needed for the machine to function are hidden in a variety of Leonardo's artworks: the maquette of the Sforza, the Da Vinci Codex, and a scale model of DaVinci's helicopter design. Sister Anna Baragli (Andie MacDowell) is an operative for a secretive Vatican counter-espionage agency, which has arranged with the CIA to assist in the Roman portion of Hawk's mission, though apparently intending all along to foil the robbery at St. Peter's Basilica.
Throughout the adventure, Hudson is foiled in attempts to drink a cappuccino. After blowing up an auctioneer to cover up the theft of the Sforza, the Mario Bros. take Hawk away in an ambulance. Hawk sticks syringes into Antony Mario's face and falls out of the ambulance on a gurney, and the Marios try to run him down with the ambulance as his gurney speeds along the highway. The brothers are killed when their driver, startled by the array of syringes in Antony's face, crashes the ambulance. Immediately afterwards, Hawk meets CIA head George Kaplan (James Coburn) and his CIA agents–Snickers (Don Harvey), Kit Kat (David Caruso), Almond Joy (Lorraine Toussaint), and Butterfinger (Andrew Bryniarski)–who take him to Darwin and Minerva Mayflower. Hawk successfully steals the Da Vinci Codex from another museum, but later refuses to steal the helicopter design. Tommy Five-Tone fakes his death so they can escape. They are discovered and attacked by the CIA Agents, and Kaplan reveals that he and his agents stole the piece, and unlike Tommy and Hudson, had no problem killing the guards. Hawk and Tommy cause Snickers and Almond Joy to be killed by their own explosive device, and they escape. Kit Kat and Butterfinger take Anna to the castle where the Macchina dell'Oro is being reconstructed.
A showdown takes place at the castle between the remaining CIA agents, the Mayflowers, and the team of Hudson, Five-Tone, and Baragli. Kit Kat and Butterfinger are betrayed and killed by Minerva. Tommy fights Darwin and Alfred inside Darwin's speeding limo, and Hudson fights George Kaplan on the roof of the castle. Kaplan topples from the castle and lands of the roof of the limo. Alfred plants a bomb in the limo and escapes with Darwin; Tommy is trapped inside and Kaplan is hanging onto the hood. The bomb detonates as the limo speeds over a cliff. Darwin and Minerva force Hawk to put together the crystal powering the machine, but Hawk intentionally leaves out one small piece. When the Mayflowers activate the machine, it malfunctions and explodes, killing Minerva and Darwin. Hawk battles Alfred, using Alfred's own blades to decapitate him. Hawk and Baragli escape the castle and discover Tommy waiting for them at a cafe, having miraculously escaped death. Hawk finally gets to enjoy a cappuccino.

Question: How do Minerva and Darwin die in the story?
Answer: The La Macchina dell'Oro malfunctions and explodes killing them