I had an interior designer we work for order walk off mats
with that same black cotton border. You know how some
of them are "it looks pretty". Well she forgot, people track
in salt and snow. So after we cleaned them it totally faded
the black dye. I took them back to the shop, hung them on
the drying rack, masked off the border and made sure to cover
the rug with newspaper. Then I went to Auto Zone and bought
2 cans of black automotive carpet spray paint, I think made by
Duplicolor. Spray painted them with 2 coats and they looked brand
new. Until more salt and snow was tracked in and it got black
dye all over the tile floor and rugs. It was a cheap attempt to
make a designer not look too stupid. I suppose it would work
in your case if the rug was kept dry. Keep in mind any loose dye
like that "rit" or what ever may be might transfer off onto other surfaces.
Its hard to make it jet black with Rit but that auto carpet paint would work.
http://www.duplicolor.com/products/vinyl.html