I just saw that your command is not complete. Try:

sudo apt-get install sqlite3 libsqlite3-dev

If still does not work, try the following:

Running the command:

 sudo apt-cache policy sqlite3

will give you the following output:

sqlite3:

  Installed: (none)

  Candidate: 3.7.9-2ubuntu1.1

  Version table:

     3.7.9-2ubuntu1.1 0

        500 http://archive.ubuntu.com/ubuntu/ precise-updates/main amd64 Packages

     3.7.9-2ubuntu1 0

        500 http://archive.ubuntu.com/ubuntu/ precise/main amd64 Packages

It is visible that the package is in archive.ubuntu.com/ubuntu

Hence, check your repositories. Open the update manager, go to settings, and after typing your password, you will see the repositories selected:

Try the options I have selected, and you should be fine.