I suggest shooting at ISO 160 (and its multiples). Noise is about as bad as it gets at low ISO's at ISO 125. See this video that compares noise at different ISO's for the Mark III: https://vimeo.com/39581579
As Policar points out, this is normal and, at reasonable exposures and bringing back the contrast in post, the noise effectively disappears. Finally, I would make sure that HTP is off, which it probably is if you're shooting at ISO 125.
Also those intermediate ISOs are pull-up/pull-down from the base ISOs, which for Canon is the even 100/200/400/etc. At least, that's how it was explained to me. So if you go for a base ISO, it'll be good, or if you go for the 1/3 under a base (such as 160/320/etc) it actually pulls it down from the base ISO so the noise will likely be a bit less than the next higher base ISO, while from the 1/3 above (125/250/etc) will actually be more likely to have higher noise & worse quality because it's boosting it off of the base ISO.
That's my understanding, someone want to correct me?