HR advice needed for small business

We have fewer than 10 employees. It has come to my attention that one of them is corresponding with potential future employers while at work, using our time/computer/network & email address! This person has signed a company "no expectation of privacy" policy memo. Our handbook doesn't specify that job-seeking is not permitted while you're on the clock, but we consider this time theft. Should this person be written up and given a suspension or can this person be fired with a degree of certainty that no unemployment compensation will be awarded?