Bricks, Masonry, Asphalt and Concrete - Can you cut out a slice of concrete?

OK..never had to do this before, so I have no idea if this is a simple thing or not. I am tryign to install some french drains around my new house to help with some drainage issues we've noticed. One of the problems is that rain empties from a downspout, and then flows across my concrete driveway until it hits a little "channel" that takes it to a sewer at the corner of my property.

My thought had been to cut a strip of concrete out of the driveway and install a drain to help collect the water. But I'm not sure if that's easy or just plain impossible to do.

Any thoughts, or alternative ideas? There isn't enough room between the driveway and our house to channel it around the concrete.

You can cut a groove out with a diamond saw, lay a pipe and patch. The problem is that you will have seriously weakened the slab uunless you take steps to tie the seperate pieces back together. If the driveway is not really wide you could try an dig under it. I have only had to get ridgid metal conduit up to about 3 inches under a driveway. You can drive it under with a sledge hammerand a garden hose as a primitive jet. Pitch was not much of an issue for this.
If you decide to cut it, I would try and drill the edges and slide some rebar in on each side to attach the patch to the existing slab.