Had the same problem and found the easy solution thanks to inspiration from the above answers. Here's a quick step-by-step summary:
First open MMC with the Certificates plug in.
Drag-n-drop your new certificate (missing the key on the upper left part of the certificate icon) to the "Personal" certificate store. This I did because the name of the "Web Hosting" store is a so called friendly name and not the real name of the store, and I could not remember the real name which is needed for the command prompt utility certutil. Instead I just remembered that the real name of "Personal" is "My". Makes the rest easy, and once done I just move the certificate back into "Web Hosting".
Once the new certificate (missing the key) is in the "Personal" store, start a command prompt and issue the following command: certutil -store "My" (assuming the quotes are needed)
Note the serial number of your certificate. It's in the first line of the certificate dump. If you have other certificates in the "My" store, then you need to find the one you just moved. Look at expiration date and name for example. Mark and copy the serial number.
Now issue the command certutil -repairstore "My" <paste serial number here> and note the private key is verified.
Move the certificate back to the "Web Hosting" store and refresh. You should now see the certificate icon overlaid with a small key icon in the upper left part.
Now you should be able to choose the certificate from inside the IIS bindings dialog.