It will, but not if you try to export it at the original number of pixels. You need to visit the 'Image Sizing' panel in the Export Dialogue and reduce the number of pixels to something that will compress to your 1000k size:
The numbers I show aren't necessarily the numbers you want, and if you specify your size in pixels, the Resolution parameter will be totally ignored. You'll need to think about what you're doing and to experiment a bit.
Thank you for the super fast response! That worked a treat.
I really appreciate your help!