You can do this by using a Bézier extrusion with animated start/end values, which was added in Blender 2.65. You'll need two curve objects: the path and the circle (if you don't have a circle curve, just add a Curve → Bezier Circle).
First, select the path, and set its Bezier Object (under the curve options tab) to the circle. This will take care of the extrusion portion. To avoid the "funky taper," just don't set the taper object.
Then, animate the End Bevel Factor value. To do this, go to frame 1, and set the End Bevel Factor value to 0. Hover over it and press I, or RMB → Insert Keyframe to insert a keyframe. Then, go to the frame where you want the transition to end, set the factor to 1, and insert a keyframe again.
This image demonstrates the two key things to set:
NB Using the circle as a curve object allows you to have a more flexible swept object. If all you need really is a circle with no endcaps, just set the Bevel Depth to a higher value than zero and set the Fill to Full instead of Half.