The problem is almost certainly an incorrect setting in your list, in the General Options page.
The "Send password reminders to, e.g. -owner address..." option should be set to No. The Yes setting is only for use with umbrella lists, and the only such mailing lists on the Sussex system are centrally managed. If you have this option set to Yes then requests for confirmation of subscription will not reach the prospective subscriber and they will never be able to subscribe.
When you change the setting to No, remember to click the button at the bottom of the General Options page.
The option is rather misleadingly labelled because it's not just for password reminders; it's also used for subscription confirmations.
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This is question number 1659, which appears in the following categories: