sp_rename leave unaltered the definition, (see sys.syscomments)

create procedure My_proc

as

select 1 as a

go

SELECT * FROM sys.syscomments

WHERE object_name(id) = 'My_proc'

go

exec sp_rename My_proc,Myproc

SELECT * FROM sys.syscomments

WHERE object_name(id) = 'Myproc'

After the rename the definition in the columnn "text" is unaltered.

The only advantage of sp_rename instead of "drop and re-create" is that you shouldn't reassign permissions on procedure.