How can I create a GIF out of a sequence of PNG files without the previous image appearing on the next frame?

I assume you're using http://www.imagemagick.org/. The -dispose command describes what is to be done with the frame after its delay (when the next frame is loaded). Info here: http://www.imagemagick.org/Usage/anim_basics/#dispose

-dispose previous

try:

convert -delay 5 -dispose previous -loop 0 Frame_7_00000*.png" animated.gif