Slow Mysql stored procedure on RDS instance

I have a problem with a Mysql stored procedure running slow on an Amazon RDS instance.

I'm currently migrating the Mysql database of a client to our own Mysql database using stored procedures. Both databases are on the same RDS instance. The instance class is 'db.m1.medium'.

One stored procedures copies approximately 2.600.000 records from one table in the client's database to 2 tables in our database.

When running this stored procedure on a local copy of our database on my laptop, it takes 1 hour 15 minutes. When running this stored procedure on the RDS instance it would take approximately 19 hours. I'm calling the stored procedure from an EC2 instance in the same availability zone.

Can I make the stored procedure running faster on the RDS instance?