The arrangement that Takkies gave would be the best option, but it might not be possible for you to wire your cables that way.
Your problem is most definitely that you've connected the telephone on the "modem" socket, and here's why:
The POTS filters are low pass filters (which blocks the higher frequency signals that the ADSL modem uses), and the Splitter filter is just a normal splitter, with a low pass filter for the "telephone" socket.
So if you connect something on the "modem" socket, it doesn't get filtered at all.
All you need to do now is to get a splitter (not a filter), and connect it to the splitter filter where your alarm is currently connected. Then you connect both your alarm and telephone to that splitter, while leaving the modem socket empty.
Remember the alarm must be upstream of the phones and connected in series.
This way the alarm always has priority even if someone leave the phone off the hook.
Are you sure about that? I was under the impression that the order doesn't make a difference.
I'm pretty sure that if you leave the phone off the hook, that the alarm won't be able to make a call, even if it upstream to the phone.