Yes, this is completely normal. Pregnancy symptoms, particularly morning sickness, tends to start subsiding as the pregnancy progresses and you approach the second trimester. The reason is because the level of pregnancy hormone, hCG, tends to level off around this time. Also, not every woman will experience horrible morning sickness during her pregnancy... it's a highly individual experience. For example, my mother only experienced a couple of episodes of nausea while she was pregnant with me (never vomited), while I vomited every half hour of every day for the first 7 months of my pregnancy (to the point of having to be hospitalized and put on IV fluids and antiemetics). As long as you're not having any abdominal pain, discharge, or bleeding (all signs of miscarriage), I would try not to worry... though I realize this is easier said than done when you're pregnant. ;)
I hope this helps!