I'm looking for flights to go to Vienna and my best option for schedule and price has me arriving in Amsterdam at 6 am and departing for Vienna at 7 am. How possible is it to make this connection? We might do carry on only, haven't decided yet. Thanks.

I'll guess you are flying from Detroit to Amsterdam? If so (arriving from the US), you won't have to go through security again, only immigration (and then customs which is usually nothing). This is a recent change; last year, the security setup was revamped at Schiphol airport, so passengers on flights from "trusted" places like the US don't need to be re-screened.

Anyway, I'd say an hour is fine if you arrive on time; you might even arrive early. If not - I assume there are later flights you would get put on if you miss your connection, assuming you are booking this all as one ticket. You can check to see what time later flights depart for Vienna, but I'd guess there are a good number of them.

I did it in 1 hour with checked baggage last July, didn't have a problem. Both my luggage and I made it to TLS. I was flying on KLM.

JFK > AMS
1 hour layover
AMS > TLS

I think you need a newer input since the Brussels incidence.
Until that time, an AMS layover from the US to another Schengen country was a piece of cake, just passport control. Last fall, it took about 20 minutes from the arrival plane door open, went through the passport control and sat down at the departure area. According to KLM, there are 4 direct flights to VIE per day from AMS. Your flight is the first one of the day. There are many more with stop over using Skyteam codeshares. So you will be get there one way or another, but you understand that you will be given whatever seat is available on alternate flights.

Well if you've got a chance to make a 1-hour connection anywhere, Schiphol is probably the best airport to take a shot at it!

At that time of day you should be OK. There is a shortage of Marechaussee, the military police who also check passports, at the moment due to heightened security on land borders. It is advised to arrive at least 4 hours before your flight if you are leaving from Ams, but for a transfer, and at that time you are probably going to be OK. If they are selling you that then it is a legal transfer and they will put you on the next available flight if you do miss it. There are worse airports than Schiphol to be stuck for a few hours if it comes to that.

Thought I would update this post-trip. Our flight got into Amsterdam about 30 minutes early and we had more than enough time to make our connection.