The forecast, the central bank's first for that period, was made in its monthly business climate report, which highlighted further weakness in the services and industrial sectors in October in the euro zone's second largest economy.
Slowing growth has forced the government to announce a new round of tax hikes and budget cuts worth 65 billion euros over five years so that its deficit-reduction targets remain within reach.
The Bank of France said the sentiment indicator for industrial activity had fallen in October to 96 from 97 in September while its index for the services sector fell to 95 from 96 the previous month.
The Bank of France said business managers in the industrial sector had reported that activity was stable but order books were down. But service providers reported slower activity.
The Bank of France's estimate for the third quarter, for which official data is due next Tuesday, is for 0.1 percent.
Statistics office INSEE has forecast French growth of 0.3 percent in the third quarter and estimated that the economy would stall in the final three months of this year.
(Reporting by Leigh Thomas; editing by Anna Willard)