In order to qualify as a proper “star” (as opposed to a brown dwarf or a stellar remnant) it needs to be producing nuclear fusion of hydrogen in its central regions. That requires a mass of at least 83 times that of Jupiter in order for gravity to compress the central regions sufficiently.
Interestingly, despite having masses far greater than any planet, the smallest possible stars are expected to have radii very similar to that of Saturn or Jupiter. That means that in a transit survey such as WASP they can look very like planets, it is only the radial-velocity follow-up observations that show that they a stellar mass.
Alexander von Boetticher et al have just announced the discovery of the smallest known star, which with a mass of 85 ± 4 Jupiters is right at the lower limit of what is possible. The mass and radius of the object, dubbed EBLM J0555−57Ab, are shown below compared to other stars and brown dwarfs, along with (red and blue lines) theoretical models for different ages.
The discovery of EBLM J0555 was a by-product of the WASP-South survey, found because of its planet-like size. In the graphic below the star is compared with Saturn and the slightly larger M-dwarf star TRAPPIST-1.