Lenses >> bodies.
Lenses last for several bodies and hold their value. The Sigma zoom is highly regarded and very fast -- especially for a zoom lens, and I think this will be a much better combination than a full frame body with a cheap and slow lens.
I'll try to make a back-of-the-envelope calculation here: To quantify the difference: a full frame sensor has roughly twice the area than a APS-C-sized sensor. That means catch twice the number of photons, and you can double the ISO with a full frame sensor at the same noise. Now enter the lenses: the f/1.8 zoom has an aperture that is roughly 4 times larger meaning it allows 4 times the number of photons to pass. Hence, even if you take into account the smaller sensor, the combination of APS-C body and f/1.8 lens collects twice the number of photons than the full frame combo.
Of course, life is never so simple, there are other aspects to consider: First of all, for full frame bodies, there exist no f/1.8 zoom lenses, the fastest money can buy is f/2.8. And good full frame lenses are typically much more expensive, e. g. CaNikon 24-70 mm f/2.8 zooms will set you back ~$1900. Fast primes are usually also in the low 4-digit range. Also, the fast f/1.8 lens will have an advantage when it comes to flash photography. The disadvantage of the fast lens is that you have to work with a shallower depth of field if are in a dark environment. I don't think this matters that much at these focal lengths (wide to normal), but it's a point to consider.
My advice: go for the APS-C body with the Sigma. Ignore the full frame offerings by Canon and Nikon -- unless you are willing to increase your budget significantly to allow for a great lens.
