How to convert a dry-laid flagstone patio to wet-laid?

I have a patio which was dry-laid with irregular flagstones. I am not really happy with the look, or the maintenance required, as I get weeds between the stones.

Is it feasible to convert this to wet-laid without ripping up the whole thing? In other words, can I do it in sections, by picking up a few stones, putting down cement, replacing the stones in the same pattern as before, and then moving on to a new section?

Thanks