The Nottingham City Council run company was set up in 2015 with the intention of tackling fuel poverty and claimed to be the first council-run energy company in the UK.
Leeds, Calderdale and Bradford Councils own not-for-profit suppliers were supplied by Robin Hood in a 'white label' arrangement.
The West Yorkshire councils will not be affected financially.
White Rose Energy was established in 2016 to help tackle fuel poverty in Yorkshire by providing a consistent and fair price for residents who do not regularly switch suppliers.
"Its tariffs have consistently been in the best value quartile of the market and less than OFGEM’s default tariff cap," Leeds City Council said in a statement.
"Many customers have saved more than £100 per year compared to not switching.
"While the sale means the end for White Rose Energy, existing customers will be offered rates that are equal to or better than their current tariffs by British Gas.
"Alternatively, customers can choose to switch to a different provider without any exit fees.
"White Rose Energy Customers will be contacted in the coming weeks with more details on moving energy supplier and don’t need to do anything in the meantime.
"It is a good consumer habit to regularly check comparison sites to ensure that you are getting the best deal for energy. There are also a number of competitive ‘green energy’ tariffs now on the market that environmentally-conscious residents might consider."
Leeds residents and tenants can access free, independent advice on fuel bills, tariffs and energy-efficiency by contacting the Green Doctor service, a trusted council partner, on 03003 033292 or 01132 380601.
Additionally, the Leeds Money Information Centre provides up to date information to help residents find free, independent and confidential advice on a range of topics at www.leedsmic.org.uk.
These include help with energy costs and utility bills, debt and money advice and other emergency help.