Unfortunately, the easiest way I have found is to install the Confluence Source Editor. Then you just need to create a template with the same name and details in the new space, and then you should copy the XHTML code from the existing page into the new one.
As far as I'm aware, there's not yet a good solution for this, but at least this will get you pretty close.
The fact you have to buy an Add-on to copy a Template from Space to Space is weak. Yes, I know you can create Global Templates, but when you have hundreds of Spaces across multiple organizations those organizations want to be able to copy their own Templates among their Spaces. We do not want the administrative overhead of making so many Templates when the Space Admins could conceivably do it themselves.
Seems everything is a damn Add-on.
If you regularly need to make your templates available in more than one space, you can also create a Global template (you'll need to be a Confluence Administrator to do this). This makes the template available in all spaces in your site.
As Stephen mentioned, there's no easy or automatic way to move a space template from one space to another.
Have a look at our Add-on Page Tree Creator. The PTC-templates are based on normal confluence pages and can be used on any space. Best practice would be to place PTC-buttons in the spaces; like that your users simply press the button, filling out the form and the page / page tree gets created.
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