This would allow the FBI to access the phone belonging to one of the shooters responsible for the San Bernardino attack that took place last year.
In the letter penned 'A message to our customers', Cook explains the specifics of the request that has been made and declares that the US Government have "asked us for something we simply do not have, and something we consider too dangerous to create."
The iPhone has a number of 'self-destruct' mechanisms built in which mean that if an incorrect passcode is entered more than ten times the key which encrypts the entire phone is destroyed.
So far the FBI have been unable to access the phone without then risking the complete destruction of all the data held inside.
To try and get around these security mechanisms the FBI have demanded that Apple create a tool which will let them access the phone.
"Specifically, the FBI wants us to make a new version of the iPhone operating system, circumventing several important security features," explains Cook.
The Apple CEO goes on to explain that, "In the wrong hands, this software — which does not exist today — would have the potential to unlock any iPhone in someone’s physical possession."
Praising the FBI for their work Cook argues that Apple has done everything in its power to help.
"We have great respect for the professionals at the FBI, and we believe their intentions are good. Up to this point, we have done everything that is both within our power and within the law to help them."
However in the last paragraph of the letter the CEO gives a damning evaluation of what this could mean for the future - calling the FBI's demands 'chilling'.
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"If the government can use the All Writs Act to make it easier to unlock your iPhone, it would have the power to reach into anyone’s device to capture their data." argues Cook.
"The government could extend this breach of privacy and demand that Apple build surveillance software to intercept your messages, access your health records or financial data, track your location, or even access your phone’s microphone or camera without your knowledge."
"Opposing this order is not something we take lightly."
This is not the first time that Apple, and indeed Tim Cook, have spoken up against the concept of creating a 'backdoor' into encrypted devices like the iPhone.
In an interview with the Telegraph, Cook said there would be 'dire consequences' if the UK government's Investigatory Powers Bill attempted to include a clause which also required the existence of a 'backdoor' through encryption.
Apple has also sent over an eight-page report outlining both the difficulties and dangers associated with the concept of adding encryption.