The process can be a little overwhelming for Mac users, especially if you have new hardware and OS X Snow Leopard or Lion.
Calibre can't convert DRM'ed books. Library books are special because of both the Adobe DRM and the time expiration date. It is further complicated because Adobe Digital Editions for Mac can't recognize the Sony ereaders as an attached device to transfer books via the Adobe software.
Do you have the Sony software installed? You need to do that and authorize your ereader. See instructions here.
http://ebookstore.sony.com/howitworks/
If your Mac default boots into 64-bit mode, then you will have some extra steps to get the Reader Library software to work and recognize your ereader too. See Post # 2 in this thread.
https://www.mobileread.com/forums/sho...d.php?t=143993
Once you sort out the above, then here is the method that I use.
1) From your library account page, download the book. It will show up in your Downloads folder with an ACSM file extension.
2) Open this file with the Sony Reader Library software. It will import the book into your library.
3) Attach your 300 to your computer via USB and it will show up as an icon in the Reader Library software on the left-hand panel of the software window.
4) Drag-and-drop the book from your Library onto the 300 icon in the left-hand panel.
5) Eject your 300 and then disconnect the USB cable.
Hope that helps!