Well, there's one thing we should consider first. In my understanding, nuclear reactions (Hydrogen fusion, for instance) in the center of stars occur based on probabilities; there's no strict mass/temperature limit for a single nuclear reaction to occur - it just becomes less and less probable as temperature decreases. What I mean is that even if the proto-star has one half of the required mass to become a star, there can still be nuclear reactions in its center - a tiny amount of them (correct me if I'm wrong).
The key concept here is that below a certain central temperature, you don't have sustained nuclear reactions (see note below). So, against all probability, a couple of Hydrogen nuclei can fuse into Helium once in a while, but it won't happen again in a long time. So you can produce Helium, but at such a low rate that it's not really perceptible (hence, the object does not shine as a star).
Think of this as the difference between a controlled nuclear reactor (some reactions occur, with an associated release of energy) and a nuclear explosion (sustain chain reactions occur, with massive amounts of energy released). That's the difference between a proper star and a massive sub-stellar object. The minimum mass for a star (about 80 Jupiter masses) is sort of a "critical mass". (However, because my note below, the transition between controlled and chain reactions is much more violent in a star that it is with nuclear reactors - but still continuous).
In the light of this, however, your (interesting) question is still valid. So what happens if we're 1kg below the necessary mass to sustain chain reactions in the core of the star? As others have said, I think stellar evolution and nucleosynthesis cannot give answers at 1kg "resolution".
But personally I don't think the necessary mass for stellar fusion is an precise number (in the same way that the Earth's atmosphere doesn't end at one specific height - it's a continuous thing). So a 1kg difference wouldn't matter, the star would shine.
Note: when I talk about sustained nuclear reactions, I'm referring to a very specific part of the P-P chain. If, by chance (ie: quantum tunneling), two protons get close enough so that the strong force kicks in, they still haven't made it, since the strong coulomb repulsion makes this "Helium-2" nucleus very unstable. Inverse beta decay must happen at precisely the instant the protons get close enough, so that one of them becomes a neutron and you have stable deuterium (Hydrogen-2). This part of the P-P chain is what regulates Helium production, since it's very unlikely. So if temperature is too low, in practice the P-P chain will never reach its endpoint of producing Helium.
