Here is a zero-cost way to fix it for many cameras, especially the Sony NEX-7.
Step 1: Up
Not all built-in flashes allow this, but many do. For example, the Sony NEX-7 shown here has a spring-loaded pop-up flash that you can point upwards by simply nudging it backwards with your finger. This simply bounces the light off the ceiling.
The main complication is that, as you can see in the photo, this will probably require you to shift your grip of the camera slightly so that your index finger is used to nudge the flash rather than to push the shutter button. Further, you probably want to think twice before using the EVF (electronic viewfinder) when nudging the flash. If you are holding the camera to your face, you'll be using your forehead as the main reflector -- which is ok, but will give a bit of a warm color tint. Colored ceilings also may impose (easily-correctable) tints.
Step 2: It Works!
It's also worth noting that this shot was taken in my daughter's bedroom which is painted "pixie dust," i.e., pink. Thus, I did have to adjust the white balance on the bounce shot, but that was quite easy.