How can I safely replace Norton with Avast (please see details inside)?

I currently have Norton Internet Security as the anti-virus product on my PC. I want to replace it with Avast, but I have the following problem: I don’t want to leave my computer unprotected even for a second, but these two products will probably clash so they can’t be running at the same on my machine. If I install Avast while Norton is running, that will be a problem. But if remove Norton first, then my computer will be unprotected while I download and install Avast.

I’ve come up with the following strategy. Can you tell me if you think it will work?

1. While Norton is active, download the Norton removal tool. Also, download Avast, but don’t run the install just yet.
2. Disconnect from the internet.
3. Run the Norton removal tool.
4. Run the Avast install.
5. Reconnect to the internet.

A possible problem with this strategy is that Norton removal and/or Avast install may require connection to the internet in order for them to work properly (something to do with product registration maybe?). Does anyone know something about this?

Your help would be much appreciated!