The period goes inside the quote mark.
"There are peculiar typographical reasons why the period and comma go inside the quotation mark in the United States. The following explanation comes from the "Frequently Asked Questions"file of alt.english.usage: "In the days when printing used raised bits of metal, "." and "," were the most delicate, and were in danger of damage (the face of the piece of type might break off from the body, or be bent or dented from above) if they had a '"' on one side and a blank space on the other. Hence the convention arose of always using '."' and ',"' rather than '".' and '",', regardless of logic." This seems to be an argument to return to something more logical, but there is little impetus to do so within the United States."[1]

1. The Guide to Grammar and Writing, sponsored by the Capital Community College Foundation