connecting cable modem wirelessly to wired network

I have a cable modem and I would like to set up a network for my
machines and give each of them Internet access through the cable modem.

All of my computers are at the other end of the house to the cable
modem. I do not wish to wire the house with ethernet cable and neither
do I want to use an entirely wireless connection (too slow for my
liking).

I would like to network my computers together using a wired gigabit
switch (enabling them to transfer data between themselves at high
speeds) but I would also like to connect that switch wirelessly to the
cable modem so that each of my computers has access to the Internet.

I assume I need some kind of wireless device plugged in to the cable
modem and another plugged in to the switch.

Can anyone enlighten me as to which products will do this job for me?

> Can anyone enlighten me as to which products will do this job for me?

A pair access points that support bridge mode or dedicated wireless
bridges.

So, something like this?

Cable Modem <-> Access Point <-> Access Point <-> Gigabit Switch <->
NICs

Which access points support bridging in that way?

> Cable Modem <-> Access Point <-> Access Point <-> Gigabit Switch <->
> NICs

Yep

> Which access points support bridging in that way?

Loads and loads.

Linksys, Dlink, Netgear etc...

Will I need a router somewhere in the network. I only have one Public
IP address, so I will need to create a private network.