Does one have to travel into central Amsterdam first, or can you take the train directly from Schiphol to The Hague?
How much is the train? This would be on a Thursday in February, mid-morning.
When we were last in Amsterdam (2004), the train ticket machines would not take our American credit card. I don't remember what we ended up doing. How does one buy a train ticket for AMS-Den Haag without a "chip" card -- does the machine actually take cash? 
Train from Amsterdam to the Hague
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>>Does one have to travel into central Amsterdam first<<
No
>> or can you take the train directly from Schiphol to The Hague?<<
Yes. Trains running down towards Rotterdam pass through both Schiphol and Den Haag (bear in mind there are a number of stations in Den Haag - the main one the through trains use is Den Haag HS).
>>How much is the train?<<
http://www.ns.nl/en/travellers/home
>>How does one buy a train ticket for AMS-Den Haag without a "chip" card -- does the machine actually take cash? <<
There is a staffed ticket counter at Schiphol.
You can travel from Schiphol to Den Haag twice every hour direct. Check prices at www.ns.nl. I think it's about 10 euros one way. At Schiphol, ticket machines take CC, some machines take cash, but you need to have coins. Or you can buy online and print your ticket yourself, and pay by CC. Sometimes cheaper too, if you know when you'll travel.
Thank you both!