A couple of suggestions based on the info you've provided.
First, you can try flame weeding. (If your liner is very shallow (sounds like it is), you may want to skip this -- or at least test a small area to see if it is going to melt the liner.) Read the directions! The objective is simply to scorch the leaves of the weeds so that it dries up and dies. You aren't trying to burn the weeds. It may take several passes over the course of a few weeks to completely kill it -- whenever you see new grass popping up, give it another scorch.
If that doesn't work, or you can't/won't try it, I'd suggest reworking the entire area. If you're only working with 1m2, it shouldn't be too bad.
- Remove the pebbles, save them in a bucket.
- Remove the liner, discard. (it probably has a lot of holes.)
- Dig out the soil under the area to a depth of 10cm. Save it to use somewhere else -- mix it into your compost pile if nothing else.
- If you get to 10cm and still find roots from the grass, keep digging!
- Is the grass is creeping in through the edges? (I.e. does this grass live in your yard, and is invading the bed?) If so, you'll need to add some heavy-duty edging along the side of the area you dug out to prevent it from creeping back in. Find some that's the depth of the area you dug out.
- Replace the soil with a 5cm layer of coarse sand, followed by a 5cm layer of 7-12mm pea stone.
- Tamp down the sand a bit (stomp on it with your feet, or use a shovel) so it is compacted.
- For an area this small, you could consider putting an extra layer of liner between the sand and the pea stone.
- Put the liner above the pea stone.
- Replace the decorative pebbles you saved in a previous step.
I will freely admit -- this is overkill, and would probably be too much work for a large area, but for a small area with such an aggressive weed, it should do the trick.