The 47-year existence of the French chanteuse Édith Piaf was a rags-to-riches tale overstuffed with glamour, miracles, drug addiction, adulterous love affairs, non-adulterous love affairs, and enough misfortune to make an ancient Greek proscenium buckle. Truth be told, it probably had too much drama for a conventional two-hour biopic. It certainly refuses to be contained within “La Vie en Rose,” Olivier Dahan’s 140-minute attempt at encapsulation.

Mr. Dahan, who wrote the screenplay and directed the film, tries to mask the script’s cavernous elisions by hopscotching between Piaf’s late phase, when she is barely ambulant and hooked on morphine, and the heady crescendo of her early career. But this back-and-forth approach mostly just makes the chronology confusing, and it doesn’t prevent the life of France’s most adored singer from coming across — as lives so often do in biopics — as a colorful timeline of loosely related incidents. Despite a wonderful, body-and-soul performance by Marion Cotillard in the lead role and a soundtrack that makes excellent use of Piaf’s recordings, you can’t help but think there’s more to a life than this.

According to legend, Piaf was born under a streetlamp, midwifed by gendarmes. “La Vie en Rose” wisely skips this episode, introducing her as a child living with her neglectful mother, a street singer, in a working-class suburb of Paris. Her father, an acrobat, discovers his daughter’s ill treatment when he returns from the circus one day, and takes her away. No award-winning parent himself, he leaves her at a bordello in Normandy and rejoins the circus. Young Edith goes blind from an eye infection, but is cared for by an affectionate prostitute named Titine (Emmanuelle Seigner) and one day miraculously regains her sight. (The first thing she sees is a butterfly.)

Against her will, she hits the road with her father. Outside the circus tent, she has her first vision of a glittering presence named St. Thérèse. Her first attempt to sing in public, an impromptu rendition of “La Marseillaise” in the street, wins over a crowd; a few years later, she is discovered by a club owner (Gérard Depardieu). She wins over le tout Paris on his stage, and a star is born.

There’s a gypsy passion to that early scene in which Edith’s father wrests her away from Titine, and no other moment in the film can match it in terms of pure, rousing melodrama. That may be because Ms. Cotillard’s post-pubescent Edith, with her Betty-Boop eyes and marionette posture, seems almost custom-designed, as the famous refrain from “Singin’ in the Rain” goes, to make ‘em laugh. It’s an interesting tension — many agree that Piaf never realized her full comic potential — but one that Ms. Cotillard seems more dedicated than the film to exploring. Like the song for which it’s named, “La Vie en Rose” is after something simpler and more direct — or, if you like, something more maudlin: the story of a woman who overcame enormous odds to become a star, only to succumb, in her fame’s waning years, to a series of knife-twists of fate.

The most devastating of these was the death of Piaf’s one true love, the prizefighter Marcel Cerdan (Jean-Pierre Martins), who was killed in a plane crash on his way to see Piaf in New York in 1949. Ms. Cotillard’s foolishly grinning Edith is in awe of him from the moment they meet, but it’s not clear what makes Marcel so special apart from Mr. Martins’s hunky good looks — it’s a Prince Charming idea of romance. The Disneyfied romance is not necessarily inaccurate (before returning to her hotel room one night with Cerdan, Piaf really did have her assistant strew the hall with rose petals), but it would mean a lot more presented in the context of her other love affairs — all those fraught relationships with talented protégés, such as Charles Aznavour and Yves Montand, who found in her both a mentor and an unbearably reckless joie de vivre.

Mr. Dahan’s film repeatedly emphasizes that Piaf’s defining fear was being alone. A more fruitful study of her life, then, might skip the grand narrative and focus instead on the embattled relationships in which she found comfort. Not Piaf in sum — Piaf in love.