Today’s younger viewers will probably know Judy Garland mostly from the unavoidable “Wizard of Oz,” though if they watch TCM they will have encountered some of her other MGM musicals, many with Mickey Rooney, or—if they’re lucky—her “comeback” film, the 1954 “A Star Is Born,” which remains the best telling of that story even in its mutilated form.

That’s probably why Tom Edge and Rupert Goold begin their adaptation of Peter Quilter’s play “End of the Rainbow” by introducing the young Garland (Darci Shaw) on the MGM lot, where she’s browbeaten into giving up an ordinary life for stardom by studio head Louis B. Mayer (Richard Cordery), is taught by her minders to keep her weight down and her spirits high via pills, and has her romantic hopes dashed by young Rooney (Gus Barry); and why they return to the teen Garland periodically to explain the early pressures that made her the harried, drug-addicted woman she’d become by her mid-forties.

But “Judy” is primarily about Garland’s last months in 1968-1969, when, as played by Renée Zellweger with the assistance of good makeup artists, she tries to lift herself from the financial doldrums by taking a gig performing at a London nightclub run by impresario Bernard Delfont (Michael Gambon)—an engagement that proves disastrous due to her failing health and emotional fragility, not to mention her fraying vocal powers. The film also deals with her final romantic fling—and last marriage—with the much younger Mickey Deans (Finn Wittrock), who tries to resuscitate her career but proves unable to do so, as well as the custody battle she endures with ex-husband Sid Luft (Rufus Sewell) over their children Lorna (Bella Ramsey) and Joey (Lewin Lloyd), whom she has to leave with their father in California when she takes the London job.

If you want a full overview of Garland’s life and career, with all its ups and downs, therefore, you are still directed to the 2001 TV miniseries based on the book by her daughter Lorna, in which Tammy Blanchard and Judy Davis shared the role of Judy in what was a straightforwardly chronological, though heavily dramatic, account. Davis gave an especially memorable performance, earning an Emmy as well as a Golden Globe and a SAG award, although she lip-synched to Garland’s original recordings.

That’s not the case here: in the way that nowadays provides proof that an actor has really made a role his own, Zellweger does her own singing (like Joachin Phoenix did when playing Johnny Cash and Rami Malek—at least partially—as Freddie Mercury). Her task is made somewhat easier by the fact that by 1968 Garland was not in the best shape, and her voice suffered; but it’s still a pretty impressive accomplishment, and the rest of her performance is excellent as well, with Zellweger capturing the intense vulnerability behind the forty-seven-year old singer’s diva persona.

An important aspect of that is Garland’s tense, frequently difficult relationship with the Brits she has to deal with during her stay. As Delfont Gambon is pretty much reduced to scowling in the background, but Jessie Buckley delivers a nuanced, touching turn as Rosalyn, the aide who must try to keep the star under control while seeing to her needs and demands, and Royce Pierreson offers a nicely laid-back one as the club’s pianist-conductor.

On the other hand, a substantial subplot involving an elderly gay couple (Andy Nyman and Daniel Cerqueira) who idolize Garland and come to all her shows—and whom she befriends and spends an evening with—comes off as a forced commentary on the singer’s special connection with a community that, at the time, was still marginalized and persecuted. The two men also play a major role in the film’s awkwardly theatrical climax, involving a performance of “Over the Rainbow” that devolves into what amounts to a singalong. It’s meant to be triumphant but instead feels bathetic.

The film is handsomely mounted—praise is due to Kave Quinn’s production design, Jany Temime’s costumes and Ole Bratt Birkeland’s cinematography, as well as Jeremy Woodhead’s makeup, and Gabriel Yared’s score is fine, though it can’t compete with the songs. Goold occasionally pushes too hard, but he and editor Melanie Ann Oliver meld the present-day and flashback episodes reasonably well.

In the end, however, it’s Zellweger’s galvanizing performance that gives “Judy” what punch it possesses. She makes Garland as tragic a figure as James Mason made Norman Maine back in 1954, though this highly speculative telling of Judy’s weeks at Delfont’s Talk of the Town has no counterpart to her Esther Blodgett among its characters.