She had been performing at a church in the picturesque village of Goudron, in the Lot region of the south-west, when she collapsed. Storms were reported in the area at the time.
Weldens, 35, suffered an apparent cardiac arrest and emergency services were unable to revive her.
She had released her first album this year and had won several awards.
She had become a singer after growing up in the circus, according to her website. Among her biggest influences was the singer-songwriter Jacques Brel. In 2016, she won the young talent award at the Jacques Brel festival.
She collapsed at about midnight on Wednesday while singing in a local festival. Investigators were looking at electrical equipment on the stage to find out what caused the apparent electrical fault.
Weldens regularly performed with a pianist and acoustic guitarist and was in the middle of a tour. She was due to give further concerts in France and Belgium in the coming months.
Although little known, her February 2017 album Le grand H de l'homme (Man with a capital M) had brought national acclaim. A review on the Mediapart website praised the "softness of the melodies" and "the femininity and masculinity" of her work.
She told Midi Libre newspaper last year: "I've always written poems, even when I was little. Then I took piano lessons as a teen. All I wanted to do was compose".
An up-and-coming French singer, Barbara Weldens, has died on stage, apparently by being electrocuted.