"It is not necessarily weak."
Sorry to say but this sounds to me like grammatical hair splitting - I would say that - without any other context - both sentences are more readily translated as "It's not weak" - and that a sentence with "必ずしも", "そんなに" or similar would be better to introduce the concept of "not necessarily" or "not that much".....
you are making an adverbial form that sometimes is used as a noun of sorts, not really a noun but more like an adverbial noun. In those cases (where you need a noun out of an adjective) you want to use the ～さ form for i-adjectives like 弱さ which is the noun you are looking for to say「弱さが嫌い」"I hate weakness" or you can use a noun with the attribute 弱い like「弱い者が嫌い」"I hate weak people"
When は is used in this kind of sentence「弱くはない」you have to take into account that は as a particle is a contextual marker, is not part of the logical structure, so what you really are doing is literally placing a contrastive は in the middle of 弱くない in order to make a contrast between 弱く and other things that what you are describing is not. 「弱くは」"as for weak"「ない」"it's not" (but it is something else, implicitly stated by は).
This sentence is sooooooo misleading: it is not necessarily weak!!!!!! IT??? I was able to understand this sentence in English, the IT and the necessarily, only after I saw the Japanese. You can't possibly expect someone at this point in their Japanese studies to understand the function of this WA. I know it because I used to expect it a long time ago and am just recapping. A more realistic, though seemingly awkward translation, would be: Weak it is not.