Oxford City Council has broken a law against housing homeless families in B&Bs for more than six weeks.
The local authority said this was due to an "unexpectedly high number" of homeless families in September and high rents charged by private landlords.
It is the first time it has broken the rule - imposed under government legislation - since 2004.
The government said the long-term use of B&Bs to house families was "both unacceptable and unlawful".
Council housing boss Mike Rowley said it was "doing everything in its power" to tackle the city's housing shortage.
Since April, the authority has recorded 128 homeless families, compared to 107 for all of last year.
A council report said it was now "almost impossible" to find properties in the privately-rented sector affordable at housing benefit levels.
It added this was "leading to many private landlords ending tenancies of people who can no longer afford their rent".
Mr Rowley said the local authority spends more than £1.4m annually on homelessness services.
But he added that Oxford is now the most expensive place to live in the country, which was having a "devastating impact" on all areas of the city, and was only likely to get worse.
He said the city had also "run out of space", with as many as 32,000 new houses needed by 2031, but only space for about 10,000.
House building in the city is being scaled back because of government cuts to council budgets, as well as proposals to sell rented council housing, Mr Rowley added.
Gavin Dick, from the National Landlords' Association, blamed the increase in homelessness on council policies, such as one preventing investors from converting properties into shared housing.
He also said licence charges imposed on landlords who rent out shared homes were being passed on to tenants.
A spokeswoman for the Department for Communities and Local Government said since 2010 it has provided more than £1bn to tackle homelessness and support vulnerable households.