I'm not sure specifically the differences between a normal Dial-up modem and an ADSL one.
Here is my problem:
I recently found an old box in the basement, which I installed Ubuntu (linux) on. I have both a NIC, and an old dial-up modem on it.
My modem that I connect to the internet with was provided by my ISP when I first joined their service, but lately it has been overheating and I've been having my connection dropped.
Seeing as how I have this linux box with a modem, I was wondering if it was possible that I could connect to my ISP using this dial-up modem.
Here's an example:
Currently: Wall (phone cord) -> ADSL Modem -> Router -> PC, PC, PC
Theoretically: Wall (phone cord) -> Linux box (as modem AND router) -> PC, PC, PC
A DSL modem and a normal POTS modem are two totally different kinds of animals, so what you plan won't work. An old style modem translates the (audible) sounds from the phone line into a bitstream over a (real or virtual) serial line, while a DSL modem works in a totally different frequency area and translates the DSL signal into an ethernet stream (encoded mostly in PPPoE).
Also, the combination of dsl modem and router will likely need considerable less energy than the PC.
Your question actually includes a serie of good question.
1- Can I use a modem designed for to go over the Plain Old Telephone Service (POTS), like a 56K modem, as an ADSL modem?
The answer to this is no. The ADSL signal and the voice signal (used by the POTS modem) are two different kind of signal and one cannot be used to transmit data over the other.
2- Can I use a linux box as an ADSL to modem bridge? (i.e. using a computer running Linux to bridge the gap between the telephone network and an internal network)
The answer to this is yes. To do it you'll need to equip the computer with an ADSL interface card. The Linux Kernel supports a number of them. Be aware that unless your Internet Service Provider (ISP) can provide you with an interface card and offer supports for Linux computer (I don't know of any that do) you'll be on your own to make it work.
3- Should I get into trying to get that working?
If you're looking for a solution with a minimum of maintenance, the answer is, in my opinion, no. It's simpler to have a dedicated router and dedicated ADSL modem. That configuration provides a point of demarcation if you have issue. Connect a computer directly to the ADSL modem, make the connection and see if you can reach the internet. If you can connect, you know the issue lies with what's behind the modem (either the router or the computer behind the router). If you cannot connect directly, then it's either the modem, the connection of the modem to the phone line or a problem at the ISP level.
A few reference on having a direct DSL connection to a Linux computer
Yes, if you use something like a Conexant ADSL PCI card.
EDIT: It seems like that card isn't really around anymore, but there are others listed in the linux compatibility database some of which are cheap on ebay.
Why not just call your ISP and get a new modem? If your modem is truly defective, the will normally replace it without cost. If they decide to charge you, consider buying a compatible modem from Ebay / Craigslist -- I wouldn't spend anymore than $10 for one.
I'm sure replacing the modem for a working, more effective model will be cheaper and easier in the long run.
That would require a ADSL interface in your Linux box. No idea if there exists any.
SmoothWall is a linux variant for creating a linux cable, adsl, or modem router. Of course you are required to have an ADSL modem, a normal modem won't do ADSL.
Dependent on the type of dsl router/modem you have you may be able to set it up as a bridged device, thus feeding the data connection into one of your network cards. The modem router would allow all traffic through it and your firewall should be based on your linux box.
Thats what I used to do in the past using a linux box with two physical network cards and then bridged them using the linux box (plus a fairly robust firewall!).