So I heard good things about the code-two software, however it still requires a cross forest migration since you cannot have the Exchange 2003 installed in the same environment with a 2013 Exchange server. So really you have three options
1. Do a double hop, i.e. install 2010 in your environment, move all the mailboxes, then uninstall 2003, install 2013 server, migrate from 2010 to 2013, remove the 2010 server
2. Stand up a new domain and use the cross forest migration methods from MS or a 3rd party product like Code Two or Lucid8's DigiScope
http://www.lucid8.com/product/digiscope.asp
3. The only other option would be to
A: Record the X500/LegacyExchangeDN information from the 2003 server
B: Ensure that your ISP will mailbag/retain undelivered email for at least 72 hours
C: Stop all mail flow and then dismount your Exchange 2003 databases to commit all logs
D: copy the EDB/STM pairs off the server and use a tool like DigiScope to validate the data is accessible
E: Remove your Exchange 2003 Server
F: Install the 2013 Server, make mailboxes for all users and then let re-enable the mail flow into the new server, test to ensure you can sent receive
G: Add in the LegacyExchangeDN information so that pre-existing calendar items work
H: Mount the offline DB's in DigiScope and connect to the 2013 server. Right click on DB and select Restore Wizard, check to ensure all mailboxes are mapped properly and if so click restore and all historical data is restored/migrated into the 2013
mailboxes. NOTE: This can take place while users are online
NOTE: you could also avoid using a 3rd party tool by first exporting/importing all mailboxes to PST and then running the steps above