On the day with the proper tide, when the faint changes of light that had snuck their way down into the corridor outside said it was just turning to night, Sirius Black neatly taped the carefully prepared non-slide surface of a potion bottle to his skin below the waist of his filthy trousers, and poured the contents of the other potion Remus had slipped him (this one with the smooth exterior, so he couldn't confuse the two even in the dark) into his mouth, and using a stone he had loosened some time ago, pounded the bottle to a dust that fell invisible to the dirt floor of the cell. His mouth becoming numb Sirius stumbled to his cot, and swallowed the adulterated Potion of Living Death. Instantly his consciousness faded, and he fell half on and half off the cot.
When his heart stopped and his brain stopped working, a light went off down at the control room of the prison: there was a change in the status of prisoner M 3145-Black, Sirius. A pair of men were broken out of the perpetual card game in the dormitory, and directed to go up to cell and deal with it. They took a burlap bag, trolley, and form ZZ-120: Notice of Death of a Prisoner. By tradition in this, if nothing else, the staff of the prison was prompt. Disposing of corpses quickly meant that whatever weird things had been done to the prisoner in life, at least the body would be disposed of before it turned into a pool of stinking slime. Cleaning that sort of mess up was ten times the work, and the smell would sometimes linger for weeks.
The guards went through their checklist: Breath-mist on the mirror? No. Response to being pricked by a needle coated in pain-inducing toxins? No. Opinion of attending authorities? Dead at 8:23 PM. Then they gingerly slipped the cooling mass into the sack, taking care not to let their hands touch the filthy body more than they had to. As they put the sack onto the trolley, and wheeled it toward the nearest exit to the disposal area, a Dementor followed them for a while. There was definitely a soul still in the prisoner they were taking through the halls, but they were proper guards; if anything was wrong at least it wasn't its fault!