The council made the decision after a review of the future of its newspaper, explaining this would be the most 'cost-effective and popular' option.
The move is understood to have angered Communities Secretary Eric Pickles, who had expected all councils to fall in line with recent guidance put out by his department.
The Code of Recommended Practice for Local Authority Publicity, published in February, advises against municipal newspapers being published more than four times a year.
It clearly states: 'They must not appear more frequently than once a quarter, must only include material that is directly related to the business, services or amenities of the authority or other local service providers and should be clearly marked as being published by the local authority.'
Despite this, Mayor Lutfur Rahman said: 'Our analysis shows that keeping the paper weekly is by far the cheapest way of communicating important news about what's happening in our borough.'
He added that the council would be bringing costs down by reducing the number of pages and removing features such as TV listings.
The council claims that it considered options to go fortnightly, monthly or quarterly, but these all proved more expensive. At a meeting on 8 June, it was agreed that the paper should continue in its current format to meet the needs of the borough's residents.
It will, however, be reduced from 50 to 48 editions a year and expects to pay for itself through ad revenue by the end of the 2011/12 financial year.