If you are C-level and asking for advice I'm guessing it is a small company. Were it a big company, there should be policies in place that draw the lines on what should or should not be done.
In a small company, there is likely an "owner" figure, who will ultimately have the decision on the matter. Personally talk to them. Otherwise, in a big company, you'd likely have to talk to the board of advisors.
First, as long as all romantic relationships are minimally stable (i.e. no one is considering a break-up over the next month, though this is a very sensitive matter to discuss), I would advise for disclosure. People should know, so there is no hearsay. You should comment on the facts now and then so people feel free to ask about any misunderstanding that may be roaming around. Imagine for example that your SO was promoted a year ago, but you've just started dating last month, people hear that you are dating and may be suspicious that the promotion from last year was already the product of her being favored by your influence.
Nobody believes that you would treat fairly and professionally someone who you meet in family gatherings, especially when harsh decisions need to be made. This is why every serious company has policies against it. So you will very likely have to transfer or dismiss this direct report of yours. Either that or step down. I recommend talking to her about the situation so you can work together with time in your favor. Everything may be fine now, but if any relevant suspicion of nepotism is reported to the board, you will be out of time to take action, and will likely be offered the opportunity to resign from your duties.
As far as your question pictures the situation, your SO should be in an okay situation. But that is also dependent on the company's culture. I've heard of a C-level that worked for a highly hierarchical company, and because of that, his son wasn't allowed to have any job in the same company, regardless of how far he would be from his father's sphere of influence.
