Not true as far as FC goes. The uniforms were designed for first contact, and then put in use on DS9. In fact, the DS9 uniforms were not made of as high a quality materials as the FC version, despite the fact that they appeared the same on screen, due to the TV budget.
I don't know, though, if the FC uniforms ended up repurposed on DS9 or not, but I'd assume so, seems like an obvious thing to do.
Mooch is right though, that was true for Generations. They had designed all-new uniforms for that film and I believe even made some, but at the last minute decided not to use them and resorted to using a mashup of TNG and DS9 uniforms that were available to them. Picard and Data had new uniforms made in the DS9 style, but Riker wore one of Sisko's uniforms (the sleeves had to be rolled up because Frakes' arms are longer than Brooks and they'd ride up) and Geordi wore O'Brien's (which is why it always appears to be two sizes too big).
I never knew there was much in the way of confusion about this, but I googled it and apparently Robert Blackman, who was the costume designer, mirsremembered it when he gave an interview a few years back. There's this pretty wellwritten fan article though that includes a quote from writer Ron Moore on why they redesigned the uniforms specifically for First Contact. Incidently, it also talks about the aborted Generations design. There's also an older interview with the BBC where Blackman specifically refers to changing this for First Contact.
thevalkyriedirective.tumblr.com/post/60452983398
BBC Online - Cult - Star Trek - Bob Blackman - Voyager�s uniforms
EDIT: The video posted above actually says exactly this