Current Approach

Export Space as XML files (Space Tools → Content Tools → Export)
Open the entities.xml file in the exported folder
1. replace all occurences of the old space key with the new one (for example: DOC16 → DOC17)
2. replace all lowercase keys (for exmplae: doc16 → doc17)
Open the exportDescriptor.properties file → replace the space key (for example: DOC16 → DOC17)
Zip the contents of the modified export folder
The upload can be done over the system console → Import Space

Old Approach (Does not work anymore)

Copy Space is available at the Overview page of the space.

Sometimes the copy space fails. Here is a list to check

Ensure that the space you want to copy provides permission to the group "Copy Space Group"
Check that there are no restricted pages, that this group cannot read
Read the log error, sometimes stuff like a German umlaut is a problem.