Critical error in final stage of website launch – URLs are BROKEN!

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Posting as an answer from the @OneTrickPony comment above:

  1. Verify that mod_rewrite is enabled on your server.
  2. Assuming that mod_rewrite is enabled, ensure that the root WordPress install directory is WordPress-writeable, or that, if it exists, the .htaccess file in the root WordPress install directory is WordPress-writeable. (See file permission scheme for WordPress, and Hardening WordPress: File Permissions.)
  3. Then, in your WordPress admin area, go to Dashboard -> Settings -> Permalinks. Click “save” (no need to change settings), which will force a flush of the WordPress URL rewrite rules, and will rewrite the mod_rewrite rules to the .htaccess file.