I found this project that uses a GSM module (scroll down to see Picture 5). It seems the module has to be bought separately. I have old GSM cell phones lying around in the house. Shouldn't these phones contain such a chip? Is it possible to remove the chip and use it on a circuit board as shown in some of the pictures in the project?
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Does connection cables exist for your phone? Or, if it is a Nokia, are there pins behind the battery?
If yes, your old phone is probably able to receive AT or FBUS command, allowing you to easily use the internal modem, for example to send/receive sms.
Look at this page for information about the FBUS protocol (for Nokias). This other page deals with AT commands to send an read SMS (This should work with most phones).
Search about this (for example on chiphacker), many people have done such things.
There also exists standalone GSM modules (example at SparkFun).