"within the hour" means within the next 60 minutes (in 60 minutes or less starting now)

"within an hour" means over the period of 60 minutes, any hour, any sixty minutes, any day
for example:
I can complete the work within an hour = I can complete the work in 60 minutes or less, today, tomorrow or at any time in the future.

If the person wants you to complete the work in the next 60 minutes, then he/she should say "within the hour or in the next hour".

within an hour = in one hour, so it is not used if the person wants something in the next 60 minutes because "within the hour" is much more specific.

If someone said, "See you within THE hour," I'd expect to see them sometime in the next 59 minutes and 59 seconds, I guess.

If they said, "See you within AN hour," I'd expect the same thing.... that I would not have to wait more than 60 minutes.

If we are talking about some other point in time, such as 'get it done tomorrow," then I think we'd choose 'within an hour.' Like this: "Start the work at 3 o'clock tomorrow afternoon please. And I want it done within an hour!" (so by 4 o'clock) But I don't see why you couldn't say, "Start at 3, and get it done within the hour please!" It would make perfect sense to me.

But I agree with Canadian David, in that,  if a specific time hadn't been chosen, then 'within AN hour' is probably the best choice. "I can start any time and get it done in/within an hour." But again, I don't really see why you couldn't say, "I can start any time and get it done within the hour." (because the start point would be assumed to be a particular time, even if we don't yet know what it is)

Then there is American Bill who feels 'within the hour' might be 'up to the ding-dong of the hour chimes.' I think I might agree with him if it happened to be five past two. I might look at my watch and feel that the workman might be done by 3. Even ten past two, and I might feel the same. But for some reason, once we are well away from two o'clock, I would go back to thinking the man meant 60 minutes. After all, if the plumber started work at 2.55, I wouldn't expect him to finish in 5 minutes.

BUT, if he happened to say, "I'll be finished ON the hour," I'd know the job would indeed be done in 5 minutes, and I wouldn't expect a large bill!

Finally,

If I say to someone, I’ll be home within an hour (It means I’ll see you in less than 60 minutes) whereas if I say, I’ll be home within the hour (it means before 'the big hand'  strikes twelve).  

i.e. I personally believe the best point of reference is TIME

Interesting question – first time I have properly thought about it.

CAVEAT:

I'm likely not part of the consensus!

As you can read there is some dissenting opinion on this.

- Within an hour is clear, that is, within 60 minutes

- I think ‘within the hour’ is likely a little subjective – it depends on the speaker’s point of reference, whether it is TIME/the clock (within the current hour), or whether it’s the speaker’s statement (within the 60 minutes from when I’ve spoken to you/within the next hour). From the comments already made it seems both are acceptable.

For many speakers/environments though, I think it directly relates to getting something done/completed before/by the TOP of the hour. Some context would likely make this clear e.g. At 4:15, stating you need something done within the hour as you have to leave the office by 5pm

I think ‘within the hour’ is a better or more common expression when making a request.  In a workplace environment, I think its point of reference is more likely to be TIME/the clock. At 2:40, stating to someone, ‘I need that phone call made within the hour!’ Be sharp, that only gives you twenty minutes! Generally, I’d say the expression is more commonly used when there is more than 30 minutes before the TOP of the hour. 

WHEREAS, ‘within an hour’,  is used to express within 60 minutes generally, and more often than not, doesn’t refer to the NEXT 60 minutes (CONTEXT DEPENDENT) e.g. You should be able to read that report within an hour (whenever that may be). There are plenty of good hospitals within an hour from here. I’m within an hour from you.

Sometimes I think how I can learn English language if native speakers don't understand each other. :-)

In Russian, you can say "до конца этого часа(until the end of the hour)" - this will mean that if I start doing something at 2:58, then it will be ready at 3:00.

If I say "в течении часа(within the hour)", it will mean that work will be done within the next 60 minutes.