Reduplication 一X X conveys 'one X after another, accumulating slowly'. Same pattern: 一天天 (day by day), 一年年 (year after year), 一步步 (step by step). The reduplicated 周 is unstressed.
usage
Often pairs with 过去了 'went by' or with verbs of waiting/changing — it foregrounds the slow passage of time, not any specific week.
