To make this application you should first get a digital copy of a passport style photograph with a white background. You can get this taken and put onto a CD at almost any Bolivian photo studio for around 15 BS.
You should then do a pre-registration through the website SEGIP, navigating through SEGIP > Extranjeria > Solicitud de Cedula de Extranjero.
Once you have done that you need to go to the SEGIP office with a print out of your pre-registration, a photocopy of your passport including the one year residency sticker, the cover letter granting your residency and a receipt of payment of 480bs to SEGIP (contact them for the account number).
In La Paz, the SEGIP office that deals with foreigners is currently located on Calle Mexico a few blocks away from Plaza Estudiente. If all your documentation is in order they will tell you to come back in one week to collect your cedula.
Many thanks to Harry of Gringo in Bolivia for sharing his notes of the Bolivian Cedula de Extranjero application process.