The short answer is that Edith Piaf probably died of cancer on October 11, 1963. Of course, as with every other facet of Piaf's tragic life, there are several different rumors and theories as to the real cause of her death.
Piaf's life was not a healthy one, by any measure. Like so many children of the street, she was an ill child. She suffered from some sort of eye disease that left her blind from the age of three until the age of seven. Some friends report that Edith spent several years in her early teens suffering from intermittent deafness, as well. Over the years, she continued to suffer various bouts of poor health, and in 1951, she was in a series of three (some say only two) car accidents that left her with broken bones and severe bruises, for which she was given morphine to ease the pain.
She quite rapidly developed an addiction to morphine, an addiction which would torment her for the rest of her life. She also struggled with alcohol dependency and friends report that she experimented with other drugs. Sometime in the 1950s, she began to develop rheumatoid arthritis, and was reportedly in constant pain, which only deepened her dependence on painkillers. Rehabilitation programs were attempted, but were not successful, and Piaf slipped back into addiction each time she exited the facilities.
In 1959, she collapsed onstage during a concert, apparently due to the onset of a liver disease of some sort. It's unclear whether this was cancer or cirrhosis or both, but it does seem that she underwent at least one surgery to assess or repair the problem. At her final concerts in early 1963, she had a visibly distended abdomen, and cancer was suspected to be the culprit at that point.
At some point in 1963, she went with her husband, Theo Sarapo, to recuperate at her villa on the French Riviera. However, her condition rapidly deteriorated and early in the morning on October 11 (or possibly very late in the evening on October 10), she passed away. The overwhelming opinion of her friends and biographers is that her death was from cancer, probably of the liver. However, Theo Sarapo's sister says that Sarapo told her that the death was more likely because of a cerebral aneurysm. No autopsy was performed, so there's no way of ever knowing for sure.
For reasons that aren't entirely clear (possibly because Piaf had stated that she wanted to die in Paris, the city where she was born and found nearly all of her success), Sarapo and a nurse either drove or hired an ambulance to bring Piaf's body back to Paris in the dark of night, and they announced her death there the next morning. She was 47 years old when she died.
Though Piaf was refused the Roman Catholic rite of burial by the archbishop of Paris, due to her unrepentant wild lifestyle, the entire city essentially shut down for her funeral, with over 100,000 people attending her burial at Pere Lachaise Cemetery in Paris. Her grave there, beside her daughter who died in toddlerhood and Sarapo himself, who died less than a decade later in a car accident, remains a pilgrimage point for fans to this day.