A related post, made me wonder how you can hide a ring from a "special one" traveling with you to a place where you will propose and yet get it legally through security or customs? You don't want the surprise to be killed by a bureaucrat.
USAToday actually wrote an article on this a while back. The key points were:
Keep it in its box, safe and secure. (It may actually make it clearer on an xray)
Avoid wrapping the box. Security may ask you to unwrap wrapped packages.
Attach a small note - eg "Engagement ring inside, please be discreet".
Put it in a clean sock or similar, as an extra visual layer of protection, but will still be easily identifiable on an xray
Put it in your carry-on (you don't want your checked luggage going missing with it)
Fill the rest of the bag with books, magazines etc - but NOT toiletries or other items that may prompt a security search
Keep an eye on security, and if they stop your bag, try and distract her with a task while they check it (get her to find something, or buy something)
In terms of dollar value being a problem, it may be worth keeping a receipt to show value in case of any doubt (some countries limit the value of gifts or items you bring in).
I would say worry only about hiding it from the recipient - keep it somewhere that person won't look - and if you happen to get a customs search, and they're clearly going to look in your bag and find it, one of two things will happen.
If your beloved is with you, ask the customs officers to give you a moment, and then propose right then and there. After you get your yes, ask the customs officers if you can show them (and your beloved) something from the bag - they will probably allow it - and get the ring. Then take your lumps with the customs guys (having the receipt with you will help), though perhaps if your now-fiancé(e) is obviously going to wear the ring back out of the country you won't have any lumps to take - and enjoy the great story the two of you will now have about where you were going to propose (top of the Eiffel tower, picnicking in the English countryside, Mount Everest base camp) and where you actually did instead (secondary inspection at whatever airport.) Can't be beat!
If your beloved is not with you because you're being dealt with separately even though you're travelling together, tell the customs officers as soon as you can that you have an engagement ring with you that you have not presented yet. Tell them whether the ring will be leaving the country or not, and as before take your lumps when it comes to duty etc. Once again the receipt will be handy. If you have to pay duty but can get it back if the ring leaves the country, you will have to tell your beloved some time after proposing but before getting to the airport "we need to stop by customs to show them the ring is leaving and to get the duty back." Shouldn't be an issue.
Security is not an issue, a ring in a pocket of a backpack or briefcase would not raise an eyebrow from the security examiners.
If you are going somewhere that absolutely requires you to declare everything in your possession to customs or somewhere that you think might do a customs search, you can always write a note to pass to the inspector while your fiancee-to-be is distracted, telling the inspector where the ring is, its value and the fact you are trying to keep it secret until the right moment. Most inspectors are human beings and likely would understand.
But most importantly, do your homework and determine if you even need to declare personal jewelry at your destination or keep that factor in mind when selecting which country to pop the question in.
I avoided this particular issue by having it made at the destination. That may not be an option for you, but it worked out for me. In any case, good luck!
...and she said yes! ...after initially thinking I was dragging her through an elaborate joke; me not realizing it was actually April Fool's Day. The lesson here is to consider more than just the ring when it comes to planning.
What surprised me about the other post is that while countries may levy duty on returning residents’ jewelry purchases, they generally allow visitors to bring in (as long as they take out) reasonable amounts of personal property. Not just jewelry, but also computers, videocameras, etc. In some cases a bond is required, or at the least serial number registration at entry and exit to show that the articles are not for re-sale.
I get the feeling the passport and residence statuses of the other poster were complex. In general I don't think the likelihood of either airport security (a ring is a pretty obvious shape on an X-ray or customs is at all high. Most of the people I have seen pulled out of the "Green Line" through customs had appliance boxes or enough luggage to live out of for a decade.