According to [temp.spec]/5:
For a given template and a given set of template-arguments,
...
an explicit specialization s
I am going to try to summarize here what I've learned through the discussion in my other answer, in the hopes of leaving a good answer to this question, rather than having the answer be buried in the comments.
The standard says
an explicit specialization shall be defined at most once in a program (according to ODR)
ODR is the One Definition Rule. You can only define each class once within a program, with an exception designed to allow a class definition to be available in each translation unit: you can define a class in different translation units as long as these different definitions are identical, character for character. The quote of the OP is part of the ODR description, follow the OP's link to see the full description.
So IMO the standard's text above means that an explicit specialization can be defined only once, but according to the ODR, and thus with the same exceptions: you can define it in a header file so it is available in multiple translation units.
Note that it is not possible to instantiate a class without its full definition (the compiler needs to know at least how many bytes to allocate for it). The same is true for a templated class, or a specialization of such a class. So it must be possible for the definition to be present in each translation unit that uses it.
This is a definition (the specialization will be instantiated), not a declaration, it probably ought to go in a specific source file (*.cpp):
template <> struct S<int>;
Note: it doesn't "hurt" to have this in every translation unit... other than the time it takes to instantiate by the compiler, and the bloat for the object files (*.o or *.obj).
This is a declaration (the specialization will not be instantiated), and is okay to put in a header file (*.h):
extern template <> struct S<int>;
The declaration requires C++11 or later.