How to prevent gaming of website rewards for new visitors

I'm about to embark on a website build where a company wants to reward new visitors with a gift. The gift has some monetary value, and I'm concerned about the site being gamed. I'm looking for ways to help reduce the chance that any one person can drain the entire gift inventory.

The plans call for an integration with Facebook, so authenticating with your FB credentials will provide at least a bit of confidence that a new visitor is actually a real person (assuming that scripting the creation of 100's of FB accounts and then authenticating with them is no simple task).

However, there is also a requirement to reward new visitors who do not have FB accounts, and this is where I'm looking for ideas. An email verification system by itself won't cut it, because it's extremely easy to obtain countless number of email address (me+1@gmail.com, me+2@gmail.com, etc). I've been told that asking for a credit card number is too much of a barrier.

Are there some fairly solid strategies or services for dealing with situations like this?

EDIT: The "gift" is virtual - like a coupon

Ultimately, this is an uphill, loosing battle. If there will be incentive to beat the system, someone will try and they will eventually succeed. (See for example: every DRM scheme ever implemented.)

That said, there are strategies to reduce the ease of gaming the system.

• I wouldn't really consider FB accounts to be that secure. The barrier to creating a new FB account is probably negligibly higher than creating a new webmail account.
• Filtering by IP address is bound to be a disaster. There may be thousands of users behind a proxy on a single IP address (cough, AOL), and a scammer could employ a botnet to distribute each account requests to a unique IP. It is likely to be more trouble than it is worth to preemptively block IPs, but you could analyze the requests later—for example, before actually sending the reward—to see if there's lots of suspicious behavior from an IP.
• Requiring a credit card number is a good start, but you've already ruled that out. Also consider that one individual can have 10 or more card numbers between actual credit cards, debit cards, and one-time-use card numbers.
• Consider sending a verification code via SMS to PSTN numbers. This will cost you some money (a few cents per message), but it also costs a scammer a decent amount of change to acquire a large number of phone numbers to receive those messages. (Depending on the value of your incentive, the cost a prepaid SIM may make it cost-prohibitive.) Of course, if a scammer already has many SMS-receiving PSTN numbers at his disposal, this won't work.