It's a bit hard to judge "profitability" without having accounting records. Many convicts were assigned as personal servants. But let's assume "profitable" means the master is gaining something of value from the assignment, and therefore the last recorded convict assignment marks the end of its "profitability".
Convicts are convicted criminals who are transported to a distant place. In Australia, this practice took place between 1788 - 1868. Although it appears Western Australia was the last colony to abolish transportation, they never practiced convict assignment, where private individuals can apply to have a convict work for them. One of the last colonies to end the practice was Van Diemen's Land (Tasmania).
Here you can browse records of convict assignment online, on the National Library of Australia or on LINC Tasmania. Convict assignments are recorded on assignment registers or assignment lists, the latest of which goes up to 1859:
The writing is in cursive and difficult to interpret, but the columns are roughly Date (1859), Location (Hobart), Name (Thomas ???)
If you're good at researching this kind of thing, you might be able to pin an exact year. My best guess is somewhere in the 1850s.