I went to one of those ink refilling places and had the black ink refilled (I have a Canon Pixma mp210).

The guy mentioned that the printer might continue to say "Low ink," but that apart from that it would work fine.

That is exactly what happened and it was working beautifully for about 2 weeks (always with the "low ink" screen).

This morning, though, it had a different screen suggesting that there was "NO black ink left" and it only gives me the option of canceling the printing job.

Any ideas?

P.s. there is still plenty of ink left.

My suggestion is to never use re-filled ink cartridges.

The ink you get is not ink that has been qualified by Canon. It can cause nozzles to block or cartridges to be not recognized by the printer.
Purchase a new Canon cartridge and see if you can save the unit.

Using re-filled ink can void any warranty or service contract you may have.