Who needs to go: The employee or the dog?

Q: “We are a privately held company. Our owner brings her dog to work with her every day. We have a new employee that has just found out—due to health-related issues—that she is allergic to the dog. The owner said she will not terminate the employee, but the employee needs to decide whether she wants to stay or not because the dog is not going anywhere. Please advise me on this issue.” – Angela, North Carolina