Answer: Most growers cut back the flower stalk (called inflorescence) after blooming to allow the plant to rest. The belief is that the next flower spike will be stronger.
You didn't mention what type of orchid you have. For phaleaenopsis orchids, some growers induce a secondary flower spike by cutting above just above dormant buds that are located below the first flower on the stem. Sometimes, a small baby plant (called a keiki) will develop instead. Leave it until roots develop and then it can be removed and potted.
I think I'd wait and see whether or not the stem produces another bloom before giving up completely on it. Leaving the stem won't hurt the plant. If it becomes crispy dry and you can break it easily apart, then remove the stem. It obviously won't bloom again.
I hope this info helps!
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