Turn off logging, switch to simple recovery, then use a SELECT INTO statement to make a copy of the table (but with bigint column instead of int). SELECT INTO is a bulk copy operation which requires minimal resources.
GO
SELECT CAST(ColumnName AS BIGINT) AS ColumnName,
OtherColumns
INTO NewTable
FROM Table1;
GO
DROP TABLE Table1;
GO
-- Use sp_rename to rename NewTable to Table1
GO
-- Re-add primary key/constraints/etc.
GO