Is it permissible to speak with a girl for the purpose of marriage once a month while studies continue, and within the bounds of proper conduct and Islam?

It appears from your question that you possess good manners and modesty. One of the established principles in Islamic law is the rule of "blocking the means" (sadd al-dhara'i'), which involves blocking paths that lead to undesirable outcomes. I fear that through conversation, you might fall into matters you do not desire, such as occupying your heart and mind with something fruitless, and perhaps your resolve might wane and your concentration during studies might weaken. If there is no necessary benefit from speaking with this girl or other non-mahram women, then the ruling is prohibition. You might not marry this woman in the future, and thus you would have fallen into something avoidable. The advice is to propose to this girl and contract marriage with her. Then, if you wish to delay the consummation of the marriage, it becomes permissible for you to speak with her. Otherwise, avoid speaking with her.