PARIS, Oct 12 (Reuters) - The French economy barely grew in the third quarter, with the service sector just about making up for weakness in manufacturing, construction and agriculture, the Bank of France estimated on Monday.

The euro zone's second-biggest economy eked out growth of only 0.1% in the third quarter compared to the previous quarter when it stagnated, the central bank said in its monthly outlook, leaving its estimate unchanged.

Its monthly survey of executives at 8,500 firms found manufacturing output kept rising in September, but by less than firms had expected.

The central bank, which uses the survey results and data to produce its growth estimates, said demand remained solid in aerospace and the electrical equipment sector, which has benefited from the data centre build-out.

Construction activity barely grew, helped by a rebound in structural work on sites delayed by this summer's heatwaves, though order books remain thin, particularly for single-family homes.

Executives reported companies were caught between rising costs and weak sales. Prices for industrial raw materials, especially energy, fuel and metals, climbed sharply in September, but firms passed on only part of the increase to customers.

Cash positions deteriorated in both industry and services, with restaurants reporting weak demand and professional services firms flagging longer payment delays.

Uncertainty among business leaders rose in September, reversing a steady decline since March, with fuel prices the main concern. The central bank said some companies are postponing projects and households are becoming more cautious with their spending.

For October, firms expect industrial output to accelerate, though car production is set to fall sharply because of temporary plant shutdowns linked to supply shortages. Service activity is expected to keep growing at a slower pace, and construction is forecast to be roughly stable.

The national statistics agency INSEE will publish its preliminary third quarter GDP reading on October 30.

(Reporting by Leigh ThomasEditing by Tomasz Janowski)

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