If she continues her defiance, Bompard is likely to face punishment as a public official who "takes steps to prevent the execution of a law".
Same-sex marriage became legal in France earlier this year despite a vociferous campaign by traditionalist Catholics and right-wing politicians.
She could face up to five years in prison and a fine of 75,000 euros for refusing to conduct the wedding of Bollène residents Amandine Gilles and Angélique Leroux Bollène or to pass the responsibility on to another councillor on the grounds that it was against her conscience.
She "maintained her position" in a meeting with regional Prefect Yannick Blanc on Monday morning, the prefecture said.
Bompard is a member of the far-right Ligue du Sud (Southern League), a split off from Marine Le Pen's Front National, whose vice-president, Florian Philippot, criticised her stance.
She "must respect the law", Philippot told the i-Télé channel, adding that "a religious argument has no validity in a secular republic".