How much does the baby weigh at this point? If you saw a "specialist" they are probably pretty accurate depending on measurements of the baby and how well it was visualized. What are your uterine measurements? Any date can be off by a couple of weeks. This is why inductions are not until 38 weeks because you could be 36 weeks or 40. This would be confusing and harder to plan. Goodluck!
All dates are estimates so they won't be able to give you an exact date. Because of the bleeding during pregnancy it is hard to know when you may have conceived. Later sonograms can be off significantly (by 1-2 weeks or more rarely 3 weeks). I would ask the doctor if they're going to go by their date or by the specialist's date. More than likely they will go by the specialist's date as they may have used better technology and have more expertise in measuring babies. How much you've gained won't matter; some women lose weight and some gain 50 lbs by the time they are full term.
Without an earlier ultrasound, from maybe 8 weeks pregnant, there is really no way to be certain from an ultrasound now. The ultrasound now could possibly date it to within about a two-week period from when you conceived, but nothing can be more accurate at this date.