What I do is the following. It results in no downtime, no data loss, and nearly instantaneous execution.
CREATE TABLE mytable_new LIKE mytable;
...possibly update the AUTO_INCREMENT of the new table...
RENAME TABLE mytable TO mytable_old, mytable_new TO mytable;
By renaming both tables in one statement, they are swapped atomically. There is no chance for any data to be written "in between" while there is no table to receive the write. If you don't do this atomically, some writes may fail.
RENAME TABLE is virtually instantaneous, no matter how large the table. You don't have to wait for data to be copied.
If the table has an auto-increment primary key, I like to make sure the new table starts with an id value greater than the current id in the old table. Do this before swapping the table names.
SELECT AUTO_INCREMENT FROM INFORMATION_SCHEMA.TABLES
WHERE TABLE_SCHEMA='mydatabase' AND TABLE_NAME='mytable';
I like to add some comfortable margin to that value. You want to make sure that the id values inserted to the old table won't exceed the value you queried from INFORMATION_SCHEMA.
Change the new table to use this new value for its next auto-increment:
ALTER TABLE mytable_new AUTO_INCREMENT=<increased value>;
Then promptly execute the RENAME TABLE to swap them. As soon as new rows are inserted to the new, empty table, it will use id values starting with the increased auto-increment value, which should still be greater than the last id inserted into the old table, if you did these steps promptly.