There is no conflict here, she indeed talks about completely different things.
The difference Arendt is talking about here is one that is crucial in modern philosophy: The difference between what we deem to be a matter of fact (i.e. true) - no matter whether correctly so or not - and that which is factual. Not reflecting or reflecting differently on that difference is probably the source of most historical and contemporary metaphysical discussions.
Arendt differentiates between a) factual truth and b) facts.
Factual truth is something in the space of reasons (Sellars) that has propositional form and can be talked about. It is not only the statement of a (observational) fact but also the judgement that endorses this fact as being the case, i.e. being "true". In two senses, it involves our capacities to relate to the world: our ability to observe/perceive what is supposed to be true and our ability to express this content. Hence, "it is established by witnesses and depends upon testimony; it exists only to the extent that it is spoken about, even if it occurs in the domain of privacy".
Whenever we talk or think about something as being the case, we have intentional relation to something using language (or at least: concepts that can be expressed in language and emerge in the context of language learning), i.e. all she is doing here is paraphrasing the Wittgensteinian argument of language being constitutionally intersubjective and bound by pragmatic relations to the world and other persons (Philosophical Investigations).
Fact on the other side is what actually is the case. Let me use an example to illustrate what "fact" is here: Take a stone in the Rocky Mountains. Let us assume it lay in a cave for some million years, untouched, unweathered. The stone does not need our "truth" for it being a stone, it just is what it is and was what it is before we were able to evaluate the truth of its being. The stone exists as a matter of fact. Hence, "all talk about [it] — all exchanges of opinion based on correct information — will contribute nothingness to [its] establishment". It is the Kantian in-itself, the Sellarsian being simpliciter, i.e. the material existence independent of our understanding and talk about it. Mind, this does not prevent us from having "correct information", i.e. real knowledge of factual truth about the stone - neither in Arendt nor in Kant or Sellars. It's just two different pairs of shoes, since facts just are and factual truth is about that what is (and aboutness is an epistemic/semantic/intentional category, not a metaphysical one).
I think it would be misguided to think even of this "World" as something absolute, especially since she allows for an intentional forming of this material world into a world fitting human needs in The Human Condition (1958). This ends up in effectively having two different kinds of concepts of "world", i.e. the "factual" and that part of the factual that constitutes an aspect of the human condition. The former can be what it is without having any impact on or relation to human life or discourse but at the same time can be annexed and transformed into the latter (both epistemically and factually).
Considering these purely exegetic points, all she is doing here is clearly stating that she is not an idealist, i.e. keeps truth, language, and consciousness apart from (albeit in relation to) the material world as it is. The philosophically interesting follow-up question is whether she should be understood as an implicit dualist (I think she is not since she is a pragmatist), but this goes beyond the scope of this question and involves an evaluation of her philosophy as a whole.