I've branded this as v. 0.9 as it requires minor editing before use. You will also need a place to publicly host the XML gadget file (read below for using Google Sites) due to it being beta.
For v. 1.0, I will aim to make it available on the public Google Sites Gadget repository ( Insert > More gadgets...) by which time it should require no editing of code, but rather be a case of simply inserting your page ID on a one-time settings panel during setup.
The Google Plus Badge gadget takes the code from the Badge Config page (link below) and wraps it in a gadget for use with Google Sites.
To use this gadget, you will need;
After downloading the XML file, please open it up in a text editor.
Search for the profile link ( https://plus.google.com/ 101916190985226380130 ) and replace the bolded ID with the ID of your own Google+ page. Save the file and upload it to your filehost/webserver/Google Sites.
In Google Sites, select Insert > More gadgets..
In the new window, select "Add gadget by URL" and input the URL to the XML file on your server (eg: http://gsites.jbcss.co.uk/plus/gplusgadget.xml)
Click OK and you will see the gadget has embedded the Google Plus Page Badge.
You may have noticed the white space under the gadget. If you hover over the +1 button you'll see why. If you want it smaller, feel free to change the dimensions in the gadget settings (after pasting the link), but be aware that it may look a bit off!
If you don't have a file host, you can host the gadget XML on Google Sites.
To host the gadget from within your Google site, you have two options:
The latter is prettier and provides more control, the former is simpler. It is up to you which method you use.
When you have uploaded the file to the location of your choice, right click on the Download link and copy the link address. It will look something like: http://jbcss.co.uk/plus/gplusgadget.xml?attredirects=0&d=1.
Remove the rubbish after xml in the link bolded above and copy the remaining URL. You can now paste that into the Add gadget by URL text field, and it will work.
The files are below. Enjoy!