Where did you purchase your Lady Dior? If you purchased it from a Dior boutique or concession, then show your receipt (or get your SA) and request another strap. They will charge you for this service, though. Your Dior SA may not order a replacement strap if you did not buy the bag from a Dior boutique.
If you bought a pre-owned version, then you may have to find a strap through Ebay. It may take a bit of patience since sellers don't often just sell the strap.