Zeref's Weaknesses.

[Nisemono96]

Weaknesses: His magic is significantly held down by the negative emotions of people as long as he bares the burden of humanity to atone for his sins. After the timeskip it's shown he no longer feels the need to bare this burden as long as the world resents him and thus resenting humanity. Immortality granted by his magic can be removed, thanks to contradictionary nature of that ability.

 

[Nisemono96]

I feel that this explanation can be simplified.

His primary weakeness that the wordy description above addresses is that basically:

As long as Zeref possesses sympathy and appreciation for the living his Death Curse based powers are nigh-uncontrollable and will randomly lash out, killing whatever maybe within range. However if Zeref ceases to care about the lives of others however, he will gain full control over his Killing Curse removing this weakeness.

As for his Immortality being removed, there has been nothing to show that it can be removed once gained, regardless of it's contradictory nature. So this little addition in the official description is false.

Why that is so is because even Mavis, another individual afflicted with the same Curse, never died when Zeref's Curse overcame her own, she just went into a state of living death, inactive and just barely living but living all the same.

Zeref's Immortality cannot be removed, but like during his most recent/last battle with Natsu it has been shown that powers of a certain type can overcome his Immortality and grant him death. What that power is, is infact the power of the Departed.

Natsu's Fire Dragon King Mode was a temporary state, where originally by utilizing the residual, inert Magical Power left by Igneel that resides within him, where a draconian tattoo appears on his arm that serves as the physical manifestation of the aforementioned Magic Power;

http://z.mfcdn.net/store/manga/246/54-464.0/compressed/l009.jpg

This Magic Power bestowed to Natsu is said to be the manifestation of Igneel's undying will. It is however, it is extremely finite, and once it runs out, the user (Natsu) can never utilize its power again.

With it, he was able to outright overcome the Death Magic afforded to Zeref by Ankhseram's Curse,

http://z.mfcdn.net/store/manga/246/54-464.0/compressed/l013.jpg

Zeref was surprised by the potency of this Mode's Flames, as his injuries were not healing as fast as they had before, causing Natsu to explain,

http://z.mfcdn.net/store/manga/246/54-464.0/compressed/l018.jpg

Which led to Zeref hypothesizing that the power of the Dead may be the key to overcoming his Immortality:

http://z.mfcdn.net/store/manga/246/54-464.0/compressed/l019.jpg

If this is true, and by all accounts it seemed to be working like it was until Happy ruined it's chances of succeeding, then Zeref is vulnerable to powers of the dead or others associated with the afterlife.