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Does the speaker of the phone works fine anyway ? Since you haven't mentioned that there's a problem with the external speaker then I'm assuming it's working fine.
I suspect it could be due to a third party music app. Perhaps a setting like "Play music through headphone only" or something ?
Boot into safe mode and then see if you're able to play music through headphone and phone's speaker withouth any problem.
Safe mode > http://talk.sonymobile.com/t5/FAQ/How-to-boot-your-phone-into-safe-mode/td-p/348008
Also check if there's any dirt or grim inside the 3.5mm connector
Did you try the steps I gave ?
Please perform them and then report . We can then begin with appropriate troubleshooting methods based on the outcomes.
Ok
Now this problem sounds like a hardware one. But we would still go with a software repair to completely rule it out.
Backup all important data of your phone ( contacts , messages, pics, vids everything on your PC) as it will erase everything in your phone.
Update / Repair your phone via PC Companion
Download PC Companion and install it. Do not connect your phone until you're asked to do so.
Launch PC Companion > Start Support zone > Start Phone/ Tablet Software update > select Repair my Phone / Tablet > Follow the on-screen instructions it shows.
Still did not fix it. It had worked before an now it's not so this is weird. Think its a hardware problem? I've had the phone for less than a month.
Since even software repair didn't fix it, it's most probably a hardware problem.
Please have your phone examined at nearest Sony service center and if it's under warranty ( and assuming there is no damage ) you're phone would either be replaced or repaired free of cost. If you bought the phone from Sony center then try to contact them and see what they say, they could even provide you with the details of authorised sony service center
For more information, contact sony in your country. http://support.sonymobile.com/global-en/contactUs/