id go by the u/s
if she said its 5 weeks then shes probably right and id take her advice on it

however further along during your pregnancy it may change and they may say insted of 13 weeks your 14 weeks or something like that

Yes, I'd go by the u/s. hCG ranges can vary a lot early on, and it's easy to have your dates be off, say you ovulated a week later (very possible, especially if you were not tracking ovulation with basal body temperatures), or maybe had late implantation (it can occur as late as 12 days past ovulation), that can throw off your dates. Generally, early dating by u/s is tough anyway because everything is so small. The main thing is if you go in for a repeat u/s and you see the progression expected for the timeframe between the 2 ultrasounds, and if your u/s due date looks off by more than a week, they will then usually move the due date.

Ya ... go by U/S...HCG is only used to ensure that it is rising appropriately or decreasing if there is a miscarrage occuring. HCG is not accurate at all to determine age of the fetus.
Even if you have low HCG's...as long as its doubling every 48hrs, its fine.
Even though your last Mestrual Period may indicate you are 6 weeks...that is only if you ovulate on the 14th day...we dont always ovulate on the 14th day...if you ovulated days before...that would make your fetus slightly older.