South West Trains says the tickets will be the cheapest in decades

The special offer is designed to boost travel at off-peak times on journeys between London and Portsmouth and London and Southampton.

Monday's first �1 offer is on a Southampton Central to London Waterloo service and SWT managing director Graham Eccles was set to be on board.

A total of 3,000 �1 seats a week will be available and can be booked online.

Passengers can book tickets through the megatrain.com website or through a telephone hotline.

Megatrain is modelled on the megabus cheap coach ticket plan initiated by SWT's parent company Stagecoach.

Brian Souter, Stagecoach Group chief executive, said: "Megabus.com stands for unrivalled value-for-money and first-class customer service.

These are the same qualities that customers will get when they travel with megatrain.com."