MELKSHAM held up the charge of Premier Division leaders Brislington at The Conigre.
The visitors were unbeaten in eight matches, but Melksham forced the early pace and strikers Gary Higdon and Dan Kovacs had early chances.
Home keeper Tom Drewitt had a couple of first-half saves to make but it was goalless at the break.
Brislington grabbed the lead within minutes of the restart as Rimmer's cross evaded everyone and went into the net.
In Jeffrey's free-kick was saved and Will Stead fired over as Melksham looked to respond.
But on 66 minutes Stu Greener's cross was handled in the penalty area and Higdon stepped to score is 20th of the season - the forth games in a row that Town had been awarded a penalty.
Melksham nearly won it in the dying moments as his 20-yard effort hit the post.
MORE FOOTBALL REPORTS, REACTION & PICTURES IN THURSDAY'S GAZETTE & HERALD AND FRIRDAY'S WILTSHIRE TIMES