Your Livebox is a wired and wireless router/modem. So you don't need anything else.
Just leave your old computer connected to it via the ethernet cable, and set up a wireless connection to it from your new laptop.
You should find that the wireless access key is printed on a label on the bottom of the Livebox.
Alternatively, if you prefer to use the Netgear router, set it up for both computers in exactly the same way.