Once you pledge allegiance to one faction, your allegiance will stay for a week. I assume that Bungie meant until the weekly reset with "for a week".
You cannot pledge allegiance to another faction during this week, you will have to wait.
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So are we in need of spending the price of allegiance every week (2500 glimmer per character?)? Or is it only when you change your allegiance you will have to spend glimmer?