Histon have all but guaranteed their Blue Square Premier play-off place and left Weymouth on the verge of relegation with victory on Tuesday.
Only an unlikely set of final-day results can deny Histon, while Weymouth need to win their remaining two games and hope for other teams to slip up.
Matthew Langston scored the only goal, an early close-range header from Gareth Gwillim's excellent free-kick.
The Terras had chances, Dominic Merella almost scoring an audacious lob.
The Terras continued to press and came close to an equaliser on the hour mark when Cameron Mawer's long-range effort was tipped over by Danny Naisbitt, Gavin Hoyte headed just over the top from the resulting corner.
Weymouth have not won in 14 Blue Square Premier games.
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