ERROR 2006 (HY000): MySQL server has gone away

I get this error when I try to source a large SQL file (a big INSERT query).

mysql>  source file.sql
ERROR 2006 (HY000): MySQL server has gone away
No connection. Trying to reconnect...
Connection id:    2
Current database: *** NONE ***

ERROR 2006 (HY000): MySQL server has gone away
No connection. Trying to reconnect...
Connection id:    3
Current database: *** NONE ***

Nothing in the table is updated. I've tried deleting and undeleting the table/database, as well as restarting MySQL. None of these things resolve the problem.

Here is my max-packet size:

+--------------------+---------+
| Variable_name      | Value   |
+--------------------+---------+
| max_allowed_packet | 1048576 |
+--------------------+---------+

Here is the file size:

$ ls -s file.sql 
79512 file.sql

When I try the other method...

$ ./mysql -u root -p my_db < file.sql
Enter password: 
ERROR 2006 (HY000) at line 1: MySQL server has gone away