If you wanted to use MailChimp, you could use my
MailChimp Integration for this. MailChimp includes the ability to edit the "welcome" e-mail sent to customer's after signing up, so you could put your coupon code there, and the customer would get it after subscribing.
It also includes the ability to show a pop-up modal for signing up, when customers first visit your site. You can edit the pop-up with some HTML code, or by editing the .tpl file.
There are other benefits for using MailChimp for your e-mail marketing, such as reducing server load from your server sending out e-mails, and better templates / campaign design, so it's worth considering if you haven't already. Their free account remains free up to 2,000 subscribers, I believe.
Feel free to take a look at the screenshots and demo site, and if you're interested let me know at
www.getclearthinking.com/contact if you have any further questions.