How can I rectify my passport error with the USA embassy?

I think this is beyond the level of general visa advice that this site can provide. I am not a lawyer, but this sounds like visa fraud. The US's laws against visa fraud are quite strict.

I don't know for sure, but common sense suggests the US government would be remiss not to keep a permanent record of an incident like this. So even if you apply again, using a new passport without the fake stamp, the embassy staff can look up your name and see what happened.

Presumably, you got a letter explaining your visa denial, which would explain any options you have. There may be some sort of appeal or waiver process, but I doubt it is something you can effectively handle by yourself - you'd probably need to hire an immigration lawyer from the US. This would be expensive.

I personally think your time and energy would be better spent figuring out how to tell your employer that you can't travel to the US.

Nate already provided a very good answer but I would like to stress another point. Focusing on the passport is the wrong way to approach this. What happened is that you were refused a visa, apparently because you were suspected of having obtained another one fraudulently. It's not merely a ”passport error”.

You definitely need to get a new passport without a fake visa in it but at this point you can expect the US to have a record of your previous failed attempt at getting a visa. I have no idea how detailed it is or how long it will be kept on file but I would expect it to specify you were suspected of fraud and to remain live for at least 5 to 10 years (in Europe, it's more likely than not to expire after some time, in the US it might very well be kept indefinitely for all I know). A new passport is not enough to erase that.

Also understand that evidence of earlier fraud (or overstay, incidentally) speaks to your character and reliability and therefore does matter even if the stamp was not a US stamp. Conversely, having stayed previously in other countries (e.g. in Europe) while respecting the conditions of your visa for those countries is generally considered a plus. Stamps and visas from other countries are therefore definitely material to a visa application.

So, as Nate explained, you need to figure out exactly where you stand legally (whether there is a ban or not, etc.) and what you can do next (appeal, etc.) to confront the issue. And even if the chances of success appear low, you really need to confront it openly because any appearance that you are trying to hide it (e.g. if you get a passport to get rid of the earlier stamp and then fail to disclose that you were refused a visa) are likely to make matters even worse.