I know we have an orchid expert... But can't remember who!

I thought it was Angievp. I tried to link her name but it didn't work. 
Eta--based on the post above mine, it did work!  
M~

It probably won't kill your chances. I spent a few years chopping the stems off as soon as they were done blooming.  Usually it will send up a new stem, but again be patient as it can take quite some time.  I actually discovered by laziness that sometimes it will send another flush of flowers.  I hadn't yet cut off the stem, and noticed a few weeks to a month later it was budding off the uncut stem.  Since then, I wait and see if it completely dies back - and probably half of the time that's what it does.  As I said, I'm no expert.  I pretty much ignore them and they do pretty darn well!   

Ummmmm...I'm no expert by any means, but I've been growing orchids for about 7+ years now. You probably have a phalaenopsis. The ice cube method will eventually kill it.

Phals have an on and off growing cycle, much like most plants. The stem is called a "spike" in orchid terms. I always cut it off as close to the base as possible. Growers have different prefrences regarding cutting back the spike. Some, like me, prefer to encourage the plant to rest so that the next blooming cycle can be stronger and produce more blooms. Besides, I have found that in some species and in some hybrids, leaving the spike encourages keikis, which is something I'm not interested in for various reasons.

Some prefer to let the spike stay until it shrivels up on its own. This way you will probably get a secondary blooming cycle but it is usually (in my experience) not as robust as the first one. Like I said, I discourage keikis (unless it is a valuable plant, in which case I want keikis).

As far as day to day care, Phals require lower light than, let's say Cattleyas. Most phals are hybridized to bloom as much as possible. Some tips re: day to day care: 1) DO NOT water down the crown. It will eventually rot the plant. 2) Water once like you really mean it (think jungle deluge) and DON'T touch it again until the potting medium is bone dry. If yours is in a pot, you can take a skewer and stick it gently inside. That way, you can periodically take the wood skewer out to gauge accurately when the inside of the potting medium is dry. Usually phals can take 1 to 2 weeks between waterings. 3) Fertilize with a well balanced orchid fertilizer. Ultra high nitrogen will induce leaf growth but no blooming. 4) I fertilize 3 weeks on 1 week off, meaning, I fertilize once x 3 times then the next watering I flush out the chemicals. 5) I use seaweed in liquid form to encourage root growth.

Your phal should rest about 2-3 months. After which time, assuming it is healthy, it should develop a new leaf (or 2). The way you can tell that the plant is healthy is that each new leaf should be larger than the ones below. If it is not, then it is deficient in nutrients and/or is otherwise unhealthy. After it develops the leaf, then you should see the initiation of a spike. If you are into aesthetics at all, as soon as the spike is largish either 1) start turning the plant towards the direction of the sun so that the spike will grow in that direction, or 2) gently secure it to a skewer so that it will grow somewhat straight.