What is the level of the employee? Is there a union in your organization? Is there a risk of race, caste, religion, sex, or language discrimination while pushing the employee out? What is the level of the superior? What was the topic of discussion? Does the history of the employee suggest similar behavior? Why the urgency? What if the employee continues to work for the organization? Is there a clause in the order or appointment letter pertaining to the removal of service with lesser notice? Is the employee not interested in working for your organization? Is employment by nature stressful? Has an inquiry been conducted? If so, what are the findings?
Without answering these questions, terminating an employee could be difficult for an HR personnel.
However, if you feel the employee should be let go, and if you are a small organization without union or labor problems, and if you are not a manufacturing unit, you can very well give one month's salary to relieve him. This is highly risky if you do not address a few of the vital questions listed above. Personally, I do not support the idea of terminating an employee just because a few questionable activities have been noticed.
As a good HR person, if your organization does not wish to employ a person, talk to him and explain the issues behind the decision to continue employment and take a resignation letter from him.