There is antimatter, but "a giant cloud" is very much the wrong description for it. So let's clear up some possible misconceptions.
Antimatter is not that unusual to find...
Sure, it annihilates rather quickly when it interacts with normal matter, but space tends to be rather, well, empty. So if you make an antimatter particle, it might be some time before it can find something to annihilate with. Moreover, plenty of normal matter begets antimatter: radioactive nuclei with too high a positron/neutron fraction can undergo positron emission, where indeed they emit an antimatter particle.
...But it should be a relatively small fraction of total matter.
The "cloud" mentioned is a region with more antimatter than you might naively expect, but not overwhelmingly more. It is a large region with thousands of star systems, all behaving normally, and, as I'll calculate below, consists almost entirely of normal matter, even excluding the stars.
Tracing the news to its source.
First, we should find the original scientific publications here. The story you linked was actually based on this 2008 NASA press release, which was in turn written about the 2008 Nature article "An asymmetric distribution of positrons in the Galactic disk revealed by $\gamma$-rays".
Of note is that this "cloud" is old news - the abnormal antimatter signal was seen decades ago. All this paper does is note that the signal is coming from a region slightly off-center from the center of the galaxy, which might provide a clue to the source (more about that later).
How much antimatter is there really?
The Nature article gives us some important facts. First, the antimatter signal of interest is the $511\ \mathrm{keV}$ line. This is the result of a positron ($\mathrm{e}^+$) annihilating with an electron ($\mathrm{e}^-$), producing two high-energy gamma-rays (photons, $\gamma$):
$$ \mathrm{e}^+ + \mathrm{e}^- \to 2\gamma. $$
Both gamma-rays have an energy of exactly $511\ \mathrm{keV}$, as can be shown from simple conservation of energy and momentum. Thus we know the antimatter is specifically some wayward positrons (anti-electrons), not whole atoms and molecules and such.
They say the size of the emission region is about $600\ \mathrm{pc}$ across,1 and so we'll take it's radius to be $R = 300\ \mathrm{pc} = 9\times10^{17}\ \mathrm{cm}$ They also tell us they see a flux of $F = 1\times10^{-3}\ \mathrm{cm^{-2} s^{-1}}$ (photons per square centimeter per second). Furthermore, they cite another article, "Spectral analysis of the Galactic $\mathrm{e}^+\mathrm{e}^-$ annihilation emission," which modeled the interstellar medium and concluded that in those conditions the average positron would wander for some time $\tau = 10^5\ \mathrm{yr} = 3\times10^{12}\ \mathrm{s}$.
Let $n$ be the number density of positrons in the "cloud." Now a volume $V = (4\pi/3) R^3$ will have $nV$ positrons in it, and there will be a gamma-ray production rate of $Q = 2nV/\tau$. If $A = \pi R^2$ is the projected area of the cloud, we expect to observe a flux of $F = Q/A = 8Rn/3\tau$. Thus we can calculate
$$ n = \frac{3\tau F}{8R} \approx 1\times10^{-9}\ \mathrm{cm^{-3}}. $$
The typical number density of hydrogen in interstellar space is something like $1\ \mathrm{cm^{-3}}$, so the positrons are about one part per billion of the interstellar material in this region.
Where are the positrons coming from?
The excitement in these articles is the asymmetric placement of the cloud, as alluded to earlier. In particular, the Nature article claims this asymmetry is the same as that observed in the distribution of X-ray binaries near the center of the galaxy. Now X-ray binaries are stellar-mass black holes that are accreting material from a companion star, resulting in a hot accretion disk and maybe some relativistic jets pointing along the axis of the disk. Such extreme conditions can produce positrons, and these can be blown out into interstellar space, where they persist for a hundred thousand years or so.
Alternative sources are known. For instance, ${}^{26}\mathrm{Al}$ is a radioactive isotope produced in some supernovae with a half-life of a bit under a million years. One way for it to decay is through positron emission:
$$ {}^{26}\mathrm{Al} \to {}^{26}\mathrm{Mg}^* + \mathrm{e}^+, $$
where the nuclear excited state ${}^{26}\mathrm{Mg}^*$ decays to the ground state ${}^{26}\mathrm{Mg}$ by emitting a $1809\ \mathrm{keV}$ photon. Indeed, the Nature article discusses removing part of the $511\ \mathrm{keV}$ signal based on what the $1809\ \mathrm{keV}$ line is telling us about the ${}^{26}\mathrm{Al}$ contribution to the positron population.
The other intriguing idea that has been discussed is that the observed gamma-ray excess is related to dark matter annihilation. That is, the current best guess we have for the nature of dark matter is that is is a species of particle that very rarely interacts with anything else, but can occasionally annihilate with itself, producing photons or possibly other particles like positrons.
In summary...
There are more positrons at the center of the galaxy than elsewhere relative to normal matter, but they are still a miniscule part of the interstellar medium. If there were an area consisting mostly of antimatter, or about equal parts matter and antimatter, either its surface or its interior would be incredibly bright, far brighter than any signal we detect.
1 Well, actually they say it has an angular full width at half maximum of $6^\circ$, and that at that distance $1^\circ$ corresponds to $100\ \mathrm{pc}$.
