Transfer in Frankfort

I assume you mean Frankfurt, as this is the Germany forum. Frankfort is in Kentucky.

I am a bit surprised that you can get a ticket with a combination of American Airlines and Lufthansa, as they are not in the same alliance. Assuming you are on one ticket, though:
You will not go through customs in Frankfurt, but in Budapest. You will have to pass immigration (passport control), though. The official minimum connection time in Frankfurt is 45 minutes; an hour is still extremely tight (especially considering a change of terminals, as American uses Terminal 2 and Lufthansa Terminal 1), but should work if the incoming flight is on time. Otherwise, you will be booked on the next available flight with free seats - not your problem, but the airlines'.

If you are on separate tickets and have checked luggage: Not a chance.

I've made a couple of 1 hour connections with Lufthansa / United in Frankfurtter :-) and I made it, but it was nerve racking and required a lot of very fast walking. It's a huge airport. I've since switched to Delta through Atlanta and Amsterdam where I get a two hour+ connection to Budapest.

If you aren't on one ticket and you miss your connection you are screwed and you will be walking home. There is usually a reason for super cheap deals.

I just looked at the AA site for IAH-BUD. It looks like BA is the partner that completes the trip. Saw no offering that involved Lufthansa.
Therefore I conclude you are looking at 2 separate bookings and the likelihood of a 1 hr transfer being successful is about the same as Tom Brady kicking 3 field goals in the Super Bowl.

Is that sufficient time?

NO!

OP must be in error and is flying United, there's no logical American flight that works here.

Maybe the flights were purchased through a 3rd party seller such as expedia, kayak, cheapoair or similar?

Or maybe they do mean Frankfort, Kentucky?

Emily, thats mean.

And the other issue is timeliness of liftoff from the US. On our last 2 trips, they were both late, making tight connections more problematic.

Paul, i just don't do the rat race any more. I need at least 2 full hours between flights and I am happier with 3.

Found this on the Lufthansa site-- looks like Lufthansa thinks it can be done--- but as noted its UA not AA that is the partner.
If OP really is on an AA/Lufthansa combo, I hope he was 'just looking' and did not buy that itinerary, as it looks improbable to be successful

LH7600

18:10
Houston - Intercontinental, TX (IAH) , Terminal E
11:15 +1 day
Frankfurt/Main International Airport (FRA) , Terminal 1

Operated by: "United Airlines"
Boeing 777-200/300

Duration: 10 h 05 min
Layover: Frankfurt/Main International Airport (FRA), 01 h 05 min
LH1338

12:20 +1 day
Frankfurt/Main International Airport (FRA) , Terminal 1
13:50 +1 day
Budapest (BUD) , Terminal 2A

Operated by: "Lufthansa"
Airbus Industrie A321

Duration: 01 h 30 min

Joe, i have done that route with a one hour lay over. I made it. Actually have done it twice. But it was foolish and desperate. One of the problems is that United has a miserable reputation for being on time. Its not a very good airline.

James, Emily is not mean. Frankfort is an important town. It is the capital of Kentucky. It's better to be absolutely sure than give a misleading information. Besides there are at least two Frankfurts in Germany.

It's not a very good airline.

If flying a US airline, Delta really is the premium airline, especially if flying internationally.

Ilja, I forgot that you can connect in Frankfort on Lufthansa on the way to Budapest. NO! Easy mistake. Sorry Emily, now you know and you won't make the same assumption again.

Tom, Delta is indeed the best. Rare I will fly anything else on a long haul.

I was being serious. I thought maybe Lufstansa was a new low cost airline that might be flying the Kentucky to Budapest route. There is more and more of this these days...

Emily is right. I was thinking the same. James and Tom. You are also right. Delta is the best, that's why I have my frequent miles with them.

Nevertheless, our poster, John deFlora, never returned so we will never know....