The business, which was run by Guy Blaskey, was set up to handle small creative duties for the media agency's clients.
It did not employ a full-time staff, but relied on freelance creative talent when the need arose for creative input. It will cease to trade in its present form, but Blaskey will continue to offer creative services when they are needed.
Nick Lockett, the chairman of BLM, said: "Flint made a profit last year, and was making a profit this year, so it's not because it was failing. We have outgrown the concept.
"We've won big accounts which need more than the occasional piece of creative. Our ambition has been elevated, and we need to focus our energy into that ambition."