I updated the php.ini and .htaccess files to allow 512 MB of memory to Wordpress and made sure that that wp-config.php file also used 512 MB. The .htaccess file at the root level is now forcing those php.ini settings to be observed recursively. I disabled all plugins in the database 'Options' table and renamed the plugins directory. To confirm that PHP has enough memory allocated, I created a temporary PHP file that had a single line: "phpinfo();". The output confirmed that 512MB were expected by PHP as the memory_limit.
Logging into the Wordpress dashboard, I receive an error that says "Fatal error: Out of memory (allocated 61865984) (tried to allocate 64 bytes)". The file specified the first time I saw the error was cache.php, which I changed so that it skipped the method allocating memory. Another file then produced the error. It seems that whenever the Wordpress dashboard allocates 59 MB of memory or more (from the error message, 61865984 bytes is 58.9 MB), PHP fails fatally.
Unfortunately, we cannot make any changes or provide you with any account-specific information over this public forum. So that the issue may be replicated as well as adjustments made, I recommend contacting technical support for any further information.