It wasn't terribly easy to do, but some research led me to the following information:

** First step
- BACKUP YOUR DATABASE!!
- Run update.php 2005/1/14
That updates your database structure to accept new information used by 4.6.1

*** Fix problem with blocks

- Add the following SQL through phpMyAdmin

ALTER TABLE blocks DROP path;
ALTER TABLE blocks ADD visibility tinyint(1) NOT NULL;
ALTER TABLE blocks ADD pages text NOT NULL;

*** Fix problem with the search

- Add the following SQL through phpMyAdmin

CREATE TABLE search_total (
word varchar(50) NOT NULL default '',
count int(10) unsigned default NULL,
PRIMARY KEY word (word)
) TYPE=MyISAM;

CREATE TABLE flood (
event varchar(64) NOT NULL default '',
hostname varchar(128) NOT NULL default '',
timestamp int(11) NOT NULL default '0'
) TYPE=MyISAM;

DROP TABLE search_index;

CREATE TABLE search_index (
word varchar(50) NOT NULL default '',
sid int(10) unsigned NOT NULL default '0',
type varchar(16) default NULL,
fromsid int(10) unsigned NOT NULL default '0',
fromtype varchar(16) default NULL,
score int(10) unsigned default NULL,
KEY sid (sid),
KEY fromsid (fromsid),
KEY word (word)
) TYPE=MyISAM;

- Go to the admin page, reset the search settings to default and run the cron.php script.