Keeps Going "Off Line"

[#1]

Sometimes when I try to go into a particular(legitimate) site through the Google page I find it's gone offline and I get a dialogue box saying:
"Dial Up Connection
Select the service you want to connect to
Connect to(then in a drop down box) Speedtouch Connection
(then three butoons)
Connect Settings Work Offline"
I'm supposed to be on Broadband.
Can somebody help please.

[#2]

You may well have some sort of Trojan dialler on your pc.....
What antivirus and antispyware protection do you have?

[#3]

Are you using a Speedtouch modem or another that connects via USB?
If you are it uses Windows Dial-up Networking to make the connection.  You should be able to alter one of the settings in there to stop it going off line.
If you go to Control Panel > Network Connections and right click on the Speedtouch connection then select Properties.  In the box that follows select the "Options" tab and in the drop down list for "Idle Time Before Hanging Up" select "Never".
Also go to Control Panel>Internet Options and select the "Connections" tab and either select "Always Dial My Default Connection" or "Dial Whenever a Network Connection is not Present" (whichever works best for you).
If you aren't using USB and connect via ethernet go to Control Panel>Internet Options and select the "Connections" tab.  Make sure "Never Dial A Connection" is selected.