approximately 1
A 6mm stone would be a.75 carat diamond.
To convert grams to carats for a ruby stone, you would multiply the weight in grams by 5. Carats = grams x 5. Thus, for an 8.5 gram ruby stone, the weight in carats would be 8.5 x 5 = 42.5 carats.
Zero carats means that there is no weight to the stone. Perhaps that the stone is not a diamond? There is no notation for a diamond equal to zero carats.
The carat is a measurement of weight of precious stones. A stone of 18 carats is twice the weight of a 9 carat stone.
This stone may be in the .50 carat range, but a gemologist can weigh the stone for you and give you the carat weight.
If the diamond is a round cut, a stone weighing a little over 3 carats would have the 9.5mm measurement at the girdle.
Any stone would be harmful
The mm size of a diamond stone relative to its weight in carats is directly related to the cut and the excellence of the cut. For example, a round brilliant stone measuring 8 mm in diameter at the girdle weighs about two carats. A different cut will have a different weight in carats.
Carats describe the weight of a stone. A jeweler can show you the scale used for this purpose.
The two measurements you give are not related, essentially. The MM measurement could be the diameter of a brilliant cut diamond, or one measurement of another shape. The carat weight you give is a weight measurement of gemstones.
A jeweller can estimate the carat size of a stone, however, to learn its true carat weight, a stone must be removed from its setting and weighed.
The metal has a karat value, meaning that it's gold; the diamond weight is measured in carats. Since the diamond stone and the metal ring are separable, anything stamped in the metal usually refers to the metal and not to the stone mounted therein.