That's a good point, actually. After a little help from Google, I can say you have a modem, rather than a router. When troubleshooting a connection, the terms modem and router are almost interchangeable; only a few cases require differentiation.

Upon inspection of the Sagem website, it would appear that you should have two cables from your modem - one to the filter to the phone line and one to your computer via USB. Note that there is nothing from the router to the power mains. Basically it means that the modem is drawing all of its operating power from the USB port. It adds portability and convenience, since you don't have to worry about that extra powerpoint, but it also means that you need a stronger DSL signal to maintain a solid connection. To get a stronger DSL signal, you either need to be put on a higher power profile at your exchange, or you need to move your house closer to the exchange - both are somewhat of a hassle.

I would always recommend a router over a modem but you could consider me an internet snob.
If you do get one, Linksys has my seal of approval. I also tend to recommend ethernet as an interface for connection over USB, because it doesn't need drivers. You can chuck an ethernet router into any PC and theoretically it should work straight away. Theoretically, your modem should be fine if you're not too far away from the exchange, but in all honesty, it's not doing you any favours.

As much as I'd like to tell you it's Windows causing the problem, I don't think it's that likely. Of course, it's always possible, and the more power to you if you install Linux!
There are two parts to every DSL connection:
- The local area connection, or the connection between the router/modem and the computer.
- The DSL connection, or the connection from your router/modem through your phone line to the ISP.
Since your connection works sometimes - ie. it has worked with your current configuration at all - it implies that your LAC (and Windows) is working fine. Disconnections are also more consistent with a DSL, or physical issue than a LAC issue.

Here are a few steps you can try to see if it improves the connection quality without speaking to idiots at tech support who don't actually know anything about the internet:
  • Make sure any phone/fax/thing that connects to your phone line in your house runs through a filter before plugging into the phone jack. If you don't have enough filters, try unplugging the devices without one temporarily to see if the connection improves.
    Diagnosis: If that fixes it, you'll need to buy more filters for your phone devices.
  • Before plugging stuff back in, try disconnecting everything from your phone line except the modem. If that doesn't work, switch the filter on the modem (if any - it doesn't need one) for another.
    Diagnosis: You have a faulty filter. Try each filter on the modem (with everything else) to see which one buggers up the connection.
  • Try modem on a different phone jack.
    Diagnosis: Faulty phone jack. (Duh ) Your phone company will probably replace a faulty jack for free.
  • You shouldn't have more than 5 devices connected into your phone line if you want reliable ADSL.
    Diagnosis: You need a splitter. It's a device that will separate the DSL and phone components of your line from each other, allowing you to have more than 5 phone devices.
  • As a quick check, you should have some lights on the modem. Specifically, we're looking for the (A)DSL light. If that turns off or flickers when you get disconnected, you definitely have a physical problem somewhere. If not, it could still be physical. It would be good if you could post a yes or no to this point.