The incredibly easy, unbelievably short answer to the question is this: either is appropriate.
The slightly longer answer may depend on your nationality. Adding an "st" to "among" and "amid" is generally regarded as a quirk of British English which does not alter the meaning of the words in any way. It's a perfectly acceptable variant. (Unless you're my third year writing lecturer and have an obsession with going through your students' manuscripts, ruthlessly crossing out these "st"s wherever they may be found.)
And now to demonstrate how little difference there is in usage:
"Don't worry," Vimes said, gesturing to the assorted faces behind him ranging in expression from infuriated to murderous, "you're amongst friends here."
"Don't worry," Vimes said, gesturing to the assorted faces behind him ranging in expression from infuriated to murderous, "you're among friends here."
Henry Watson Fowler's Dictionary of Modern English Usage1 states that there is "no broad distinction either in meaning or in usage between the two" and quotes the Oxford English Dictionary on "amongst" thusly:
amongst ... [generally implies] dispersion, intermixture and shifting position
Although, according to Fowler, not even the Oxford English Dictionary's own quotes really back up this statement.
Fowler goes on to suggest that "amongst" may have prevailed, rather than being superceded by "among", purely out of a desire for euphony. So if you prefer the sound of "amongst" or "amidst", then feel free to use them.
Even in areas where "amongst/amidst" is rarely used, the terms sometimes prevail in expressions or phrases. Think of things such as "talk amongst yourselves" or "amidst all the confusion", for example - even if you wouldn't normally add the "st", it can often seem necessary in these phrases.
Adding the "st" may also help you to establish a character's nationality. "Amongst" and "amidst" are generally regarded as being more English, whereas an American character would be more likely to use "among" and "amid".
In essence, any time is an appropriate time to use either "amongst" or "among" or "amidst" or "amid".
1. A Dictionary of Modern English Usage
The Columbia Guide to Standard American English - amid/amidst
The Columbia Guide to Standard American English - among/amongst