Switching contact from SMS to iMessage

So my girlfriend finally got an iMessage capable phone, but Messages on my phone still insisted on sending SMSes. Even after starting to receive iMessages in the same conversation. Even after a message sent from the Messages app on OS X was duly noted as being an iMessage!

Various attempts to fix this state of affairs had no effect: changing the contact number to iPhone, deleting the conversation(s), twiddling with Messsages settings on both phones, including the "Send as SMS" preference.

What did work was performing a reset of the Network Settings.