Gosh...getting technical for me........!
OK. I downloaded the prox sensor finder and uncovered it read 8.0 cm. Covered went to 0.0 so I assume that the sensor is OK. So what on earth can be wrong??????
Thank you sooooh much for the advice.
Well, yes then, the sensor is working properly. I don't know why some consider 3cm to be the threshold distance and others use any number of other distances, but 8cm is a good bit farther than 3cm, so if anything I would think it might blank out too soon (such as if you were trying to shade the screen from sun for instance), rather than not at all.
I can't troubleshoot it more from here than I have already, but since this is a computer and since any number of apps can influence how the computer responds to sensor input, it could easily be an app. Do you have anything installed that is perhaps designed to 'keep screen awake'? Often times things like navigation, video players, craigslist apps, any number of other types of apps which require extended viewing time will essentially turn off the sensor and interrupt the timeout schedule so the app doesn't go black while you are using it.
I would go through all your user-installed apps first and check each one's settings to see if a stay awake setting is checked. Then go through the factory installed ones to see if the same are in those. Aside from that, and this may sound stupid but if you have a screen protector on the phone it can interfere with the sensor working properly. I left the one on the phone that came on it and after about 2 weeks it wouldn't turn on when in a call...power button, volume controls, nothing worked to wake up the screen. I couldn't change it from earpiece to speaker or turn on Bluetooth, I couldn't get the dialpad to select menu prompts, I couldn't even end a call. The only thing that worked was once the other party hung up the screen would wake again.
Once I cut out a section of the screen protector over the sensor it worked perfectly. This can happen with cases too but I know you tried it without the case. You should look to see if the screen protector is over the sensor.
Finally you may be forced to try a FDR (Factory Data Reset), but first I would like you to try the phone in safe mode and see if the same thing happens while in a call.
First, power down the phone by pressing and holding the power button while the screen is on. A menu will pop up and you will select "Power off". Then confirm in the next prompt.
Then press and hold the power button until the phone vibrates shortly and release power button. Next, as soon as you see the Samsung Galaxy S4 logo, press and hold the Volume Down button. Continue holding that Volume down button until the screen awakes and you'll see a black box with white letters showing at the lower left corner of the screen saying "Safe mode". In this mode, the phone will only run those apps which are set to run from those pre-installed. It will not run any of the user-installed apps, and they won't even be in your screen positions or in the app tray.
Now try the phone calls and see if the problem persists. If it doesn't that confirms a third party app's the culprit. If the problem persists, it may be time for an FDR, and if that doesn't work...it's a lemon IMHO.
To get out of Safe mode, simply reboot the phone.
Sorry for the edit to the instructions above, my original source was incorrect. I've tested these instructions on a S4 that I have laying around and they work properly to invoke Safe mode.