Which organ develops and is fully functional in the first trimester?

The heart

Explanation:

Week 3- Gestational Age (Development of Your Baby- Week 1)

The embryo is going through a lot of fundamental growth right now, starting with the first steps of the brain, spinal cord, heart and gastrointestinal tract.

Week 4 and 5- Gestational Age (Development of Your Baby-Weeks 2 and 3)

Outbreaks of arms and legs are visible, but not clearly distinguishable. The heart is already beating at a constant rate. The placenta begins to form and is producing some important hormones, including hCG. There is rudimentary blood movement through the main vessels. The first structures that will become the eyes and ears are forming. The embryo is ¼ inch long at the end of these weeks.