Violating company policy can be the basis for a behavior-related termination. When it comes to notifying the workplace about the termination, much depends on the specific situation.
Notifying the workplace immediately is usually a good idea. Besides the staff who will deal directly with the termination (likely the IT department and the benefits administrator), other staff and regular customers should be informed that the employee was terminated for violating company policy. This is easily accomplished with an email or a letter to the staff. Do this before word gets out on its own to cut down on the spread of rumors or misinformation.
Whether or not you cite the violation depends on the nature of your business and the nature of the violation. The upside of citing the violation is that it shows employees that this particular violation is one the company takes seriously. By communicating that a specific employee was terminated for a specific action, you are effectively warning current employees that said actions will not be tolerated.
The downside of citing the violation is that you could be exposing potential loopholes in company policy. Also, you could be fueling the fire by bringing attention to an act that is harmful to the company.
For example, if an employee is terminated for taking cash from the register, it is probably best to cite the violation of cash handling policy. In this way, the situation is made clear to the workforce, and the importance of cash handling policy is reiterated. However, if a termination may result in further legal action, or if you believe that it would be harmful to your company to expose how an employee violated policy, then it is best to stick to a non-specified statement such as, was terminated for a violation of company policy.