The library was due to close in March 2013, but that was delayed after residents campaigned to keep it open.
In 2014, the Friends of Anlaby Park Library were given the lease to take it over.
Hull City Council said the CAT would allow volunteers to continue providing a community-focused base for socialising, learning, activities and creativity.
It also means the library could secure external funding to continue to upgrade the building, the authority added.
Ashton said: “What began as a campaign to save our much-loved community facility 12 years ago, has evolved into a successful and sustainable charity run by local people for local people.”
He added: “The trustees are now looking forward to building on the charity’s achievements to date and working with partners, volunteers, local community and most importantly, external funding opportunities to realise the full potential of the library.”