Does Sigma's Mount Conversion work for DC lenses to DG Lens Mount Conversion? Will my Sigma 17-70mm f2.8-f4 OS work with Full Frame Camera 6D after conversion?
No. It's not a mount issue—it's a glass issue.
Sigma's mount conversion is to change between different mount systems so it will physically mount and electronically communicate to the system of your choice. Canon EF and EF-S lenses are actually still in the same mount/signaling system. And chances are good your 17-70/2.8-4 will actually mount and communicate correctly on a 6D, since 3rd party crop lenses, as long as the back element doesn't protrude enough to collide with the mirror, can still mount and be used on full-frame cameras, since they usually don't include the physical "bumper" that Canon puts on their EF‑S lenses that prevent them from mounting on full frame bodies.
But optically, it will still only be able to project an image circle large enough to cover an APS-C sensor. To increase the size of the image circle, the entire lens design has to change. This is the main significance of the DC/DG designations.
If you read the text on the Sigma page for the 17-40/2.8-4 DC Macro HSM, you'll see the following mount conversion note (bold for emphasis is mine):
SIGMA MOUNT CONVERSION SERVICE
Sigma will convert the current mount of your Sigma Global Vision lens to another of your choice. Sigma lenses currently from the Contemporary, Art or Sports line, in Sigma, Canon, Nikon, Sony, Pentax mounts can be converted to Sigma, Canon, Nikon, Sony or Pentax. They cannot be converted to either an APS-C or a Full Frame format.