Determining the date of conception is typically done in one of two different ways, either through counting backwards from the last menstrual period or through ultrasound
dating based on specific features of the fetus associated with a specific stage of development.
Going by the date of your last menstrual period, your estimated due date would be August 29, 2013 with a date of conception of December 6, 2012.
Going by your ultrasound date, the date of conception would be December 8, 2012.
Both the last menstrual period and ultrasound dating are both estimates, not iron-clad guarantees of certainty, so it is impossible to say with 100% certainty exactly when conception occurred. However, given that both methods are placing a date of conception in early December, it seems statistically most likely that you conceived on December 5th.
It sounds like you are already taking steps to help with medical management of your pregnancy in getting an ultrasound. It is important to make sure that you have regular visits with your obstetrician during pregnancy to make sure you stay in good health during your pregnancy and have the best possible outcome. Your doctor
can also answer any other questions you may have about pregnancy.