The Wikipedia page about integrity is useful here. It defines the word this way:
a concept of consistency of actions, values, methods, measures, principles, expectations, and outcomes. In ethics, integrity is regarded as the honesty and truthfulness or accuracy of one's actions. Integrity can be regarded as the opposite of hypocrisy, in that it regards internal consistency as a virtue, and suggests that parties holding apparently conflicting values should account for the discrepancy or alter their beliefs.
To summarize (and simplify a bit): honesty is about what you say; integrity is about what you do. In your example, yes, the employee demonstrates both honesty and integrity, but the act of being meticulous about reporting his hours (assuming that doing so is, in fact, consistent with his espoused value system) is an excellent example of integrity.
Now, the above definition is essentially an ethicist's version of integrity. If the employee lied on his time report, but in so doing he remained perfectly consistent with his own internally consistent value system, most ethicists would maintain that he was acting with integrity—though not with honesty, of course. On the other hand, as the Wikipedia article states,
In common public usage, people sometimes use the word "integrity" in reference to a single "absolute" morality rather than in reference to the assumptions of the value system in question. In an absolute context, the word "integrity" conveys no meaning between people with differing definitions of absolute morality, and becomes nothing more than a vague assertion of perceived political correctness or popularity, similar to using terms such as "good" or "ethical" in a moralistic context.
In this sense, the employee would most likely (barring any number of vague hypothetical situations involving the "greater good") be said to have neither honesty nor integrity if he lied on his time sheet, no matter how consistent his personal views on the matter.