How to change primary key's data type from int to bigint in a huge transactional database under tremendous load?

This is a legacy system, with more than 1 billion records in a table that undergoes 10 million transactions per day.

Now the application complains that:

Arithmetic overflow error converting IDENTITY to data type int

We want to change that id column's data type to bigint, but it can't do it and times out.

What should we do? I have no clue at all. We can't stop the system, because it's a monolithic application and database and does many things. Thus we prefer not to stop the whole system.

We're using SQL Server 10.50.6000.34, that is SQL Server 2008 R2 SP3 (September 2014)