Celtic had an afternoon to forget as they finished with 10 men and 10 points adrift of Rangers in the Scottish Premier League table after a 2-0 defeat against Hearts at Tynecastle.
Rudi Skacel claimed his first goal of the season with a drilled left-foot finish before Kris Commons was shown a straight red card for an ugly challenge on Adrian Mrowiec and Ryan Stevenson duly wrapped it up.
Fraser Forster had made a fine early save from Stevenson, who had latched on to an Ian Black free-kick with the visiting defence dozing before Hearts skipper Marius Zaliukas hit the post.
Jamie MacDonald then did well to keep out Mohamed Bangura's close-range header and the Hearts keeper twice foiled Commons as Celtic rallied.
Hamill's powerful free-kick had Forster at full stretch before Skacel pounced on 58 minutes to break the deadlock.
And when David Templeton teed up Stevenson nine minutes from time he thumped a shot past the exposed Forster to send the home fans into raptures.
Bhoys boss Neil Lennon revealed his feelings about his side's title hopes after the defeat.
"It is a concern, obviously," said the Parkhead boss.
"We have a game in hand but we have to win that game and cut it to seven and eat away at it.
"But we have to find more consistency in our performances. We have lost three games already this season and last season we lost four.
"It is a difficult position that I am in but I understand that position.
"The players are going to have to turn it around, and I am going to have to turn it around."
On Commons' dismissal, he said: "One player lost his discipline and let us down badly. I didn't get a good look at it but from what I can gather it was a lunge and it is high.
"He came in and held his hand up but it is too late."