You should discuss the matter with the employee in private to determine why he or she feels they must leave. Ask whether the issue is employment related and if anything could be done to increase their chances of staying on.
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However, if the decision is final then you need to accept it. Thank them for their work, wish them well and provide them with a good reference, if appropriate, for their next employer.
After the staff member leaves, you need to start thinking about whether or not you need to replace them. If their workload can be spread across other employees without reducing productivity, consider spreading their responsibilities among other staff.
If the workload is too much to distribute, start looking for a new staff member. Create a detailed job description and set up interviews as quickly as possible.
Also consider how you will tell other staff the employee has left. There is no reason to divulge particularly private details – simply say the employee is pursuing new opportunities, and the business will continue as it always has. Morale can quickly take a dive when staff leave, so keep spirits high and remind employees of the vision you have for the business.