Not sure about the viability of that route, but it is cheap - and not too long - to travel Norwich-London-Southampton by train. Indeed, the Norwich-London Liverpool Street sector costs from just £5 - and when I travel the route, which is several times per week, I get it at that price 9 times out of 10. (The other 1 time is at £10 - the second-cheapest price.) It takes just under 2 hours and operates every half-hour on weekdays (hourly at particular times).
From Liverpool Street, simply transfer by tube to Waterloo, thence board a Southampton-bound train. It takes about 1 1/4 hours to Southampton and costs from £16.00 one-way - expensive.
So, you could get Norwich-London-Southampton for £27 one-way including 1x £3 tube ticket, and it should take, centre-to-centre, about 3.75 hours. Not bad.
Or as a London-Southampton-London ticket costs from £16.75 - just 75p more than a darn one-way ticket - you could get Norwich-London-Southampton-London-Norwich for £32.75 - including 2x £3 tube tickets.
[Edited 2006-07-21 16:19:29]
"Everyone writing for the Telegraph knows that the way to grab eyeballs is with Ryanair and/or sex."