you need to click the Day/Month/Year dropdown selection arrows and choose the blank values.
Thank you for your quick reply. I had tried that bit of editing and saved the edit, but the birthday did not disappear. I uninstalled Skype, waited a week, and reinstalled it. There was the birthdate again! Now since posting this, my birthday has disappeared from the online profile mysterious, without my doing anything. Wonder if my old edit somehow caught up.