In particular this method also works in Android 4.1.2 so it's not duty to install 4.2.2 .
Attention: i'm new user so i can't post download link, google and if you don't found it send me a PM and i will send you the proper download link
1) Install Android 4.2.2 Jelly Bean S3 I9300 for downloading it search on google: I9300XXUFME7_I9300OXAFME7_OXA.zip
2) For Rooting Samsung S3 I9300 you need:
2.1)Odin3-v1.85 search on google: Odin3-v1.85.zip
2.2) CF-Root-SGS3-v6.2 search on google: CF-Root-SGS3-v6.2.tar
3) Modem DDEMG2 And Modem XXDME4 search on google : modem_XXDME4__.zip and Modem+DDEMG2.zip
1) install Android jelly bean 4.2.2 with odin
2) root your phone with CF-ROOT
3)move the two file Modem DDEMG2 and MODEM XXDME4 to the SdCard.
4)Turn your Mobile in Recovery Mode ( Volume + + Home + Power )
5)choose zip from sdcard
6) choose the 1st zip file : modem_XXDME4 and install it
7)reboot system now
8)turn your mobile again in recovery Mode choose zip from sd card and now choose the 2st zip file: modem DDEMG2 and install it
9) reboot your phone
10) wait a bit then dial #*06# and check that your IMEI number was restored to the original one!
Hope this method also work for you.
For any Question write me.