The 25ft-long (7.6m) three-tonne car, nicknamed the "fastest kettle in the world" and the "lean, green, mean machine", has been 10 years in the making in Lymington.
It is being driven by its main financial backer Charles Burnett III.
But project manager Matt Candy earlier admitted there had been problems during the test runs.
"We've had overheating electrical components, so we've started using an awful lot of dry ice every day to cool those," he said.
"We've also had a lot of gas pressure problems, due to the temperature causing the liquid propane to vaporise further up the car than it normally does.
"We've had vapour locks. We've had pumps that couldn't cope with the extra pressure."
Fred Marriot set the record in a steam car dubbed "the Rocket" in 1906
But the car still achieved a 137mph (221km/h) average over two test runs.
The British team includes test driver Don Wales, nephew of the late Donald Campbell and grandson of Sir Malcolm Campbell. Between them, those two ancestors achieved more than 20 land and water speed records.
Mr Wales told the BBC that the main engineering obstacle was to develop a compact boiler system which every minute could turn 40 litres (8.8 gallons) of water into superheated steam at 400C (752F), at 40 times atmospheric pressure.
The team was forced to abandon a test run in March on Thorney Island, near Emsworth, when technical problems set in.
The longest-standing land speed record is recognised by the Federation International Automobile (FIA).
American Fred Marriott set the record in a steam car dubbed "the Rocket" on Ormond Beach, Florida, now known as Daytona Beach.
Usain Bolt set a new world record of 9.58 seconds in the men's 100m at the 2009 World Championships in Berlin, a speed of 23.3 mph.
Mark Webber had the highest average speed of 119.61 mph in his Red Bull at F1's Hungarian Grand Prix in July.
Mallard, a Class A4 steam locomotive, set the steam locomotion land speed record - 125.88 mph - on 3 July 1938.
Fred Marriott has held the current land speed record for steam-powered vehicles since 1906, when he reached 127.659 mph in his Stanley car.
The British Steam Car team say their car has reached 137.14 mph in testing, but this is an unofficial figure. They hope to reach up to 170 mph in their record attempt (indicated by yellow bar).
British-built Thrust SSC set the world land speed record in October 1997, reaching 763.035 mph in a desert in the US state of Nevada.
The speed of sound changes according to factors including temperature and altitude, but a standard figure given is roughly 768 mph.
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