This build is designed to give a whole lot of survivability while still being able to do a lot of damage. It is a little tricky to use well though but using this build I can still get 10+ kills a game, and once I went 17 kills to no deaths using it.

The runes are geared for burst damage (ability damage) off the start so you can get your items faster. These are very easily changed around for ability power per level runes if you would rather have them for burst damage since they are better after level 9ish. You can also switch some around for armor if you find it too hard to survive long enough to get the expensive items, or mana/mana regen if you are running out too fast.

The masteries are also geared for burst damage, but also a bit of mana/mana regen since Anivia is very mana heavy (relies on it to do most of her damage). Again this is geared to get more money at the start of the game off of minions or even champion kills. If you find that you cannot survive long enough to get some of the armor items fast enough to be of much help to your team before mid-late game, consider taking down some of the utility or offense runes and bring them into armor and/or magic res

As you can see the items are really what makes this build surprisingly effective. I only showed the items to build to what it looks like by the end. The Sapphire Crystal is an essential starting item since it gives Anivia an extra ability or two before she runs out of mana. From that I just showed the complete items since I have built these items in numerous different orders and it still worked fine. The Ninja Tabi's 10% damage reduction is very helpful and it is one of the cheaper boots so you can easily get it in one go if you attack the enemy minions or champs from a distance. The Rod of Ages at max stacks is very good for survivability and therefore should be the first big item you get. After this point the rest of the build order is very interchangeable and should be built in accordance with what your dealing with. If you want to deal with the melee champs better then the next item should be Zhonya's Hourglass or Thornmail. If it is a melee carry you wanna deal with faster then Thornmail would be a great item after the Rod of Ages. If the mages or ability power champions are being a pain then the Force of Nature is a good one to get next. If you just want more health then Rylai's Crystal Scepter is a great fourth item, but other then that it should be one of the last items you get if not the last item you get. If it gets to late game and you find yourself doing rather well, sell the Ninja Tabi and get a Warmog's Armor for over 3000hp

The skill sequence is once again more designed to be offensive considering how Crystallize is level 1 for a very long time. Flash Frost has that stun that is possible to cut off early to hit and stun closer enemy champions, in which case popping up even level 1 Crystallize after they are stunned will still slow them down. As most builds, getting Glacial Storm and upgrading it as soon as possible is very helpful in both offense and defense, though more offense since it slows as long as the enemy champion(s) are in it. An example of a combo where, even at level 7, is effective is; Lead off with a Flash Frost and let it go very slightly past the target enemy or enemies and cut it off for Flash Frost's double hit. Once they are stunned, throw off a Frostbite and then immediately pop Glacial Storm leaving a little space between where the enemies are and the edge of Glacial Storm then pop off a Crystallize blocking the quickest way for enemy champions to leave the storm. Even at level 1, Crystallize is a big enough wall to force the enemies to stay in the storm for a little longer, which can lead to you getting a kill or teammates getting kills.

Simply, Flash is one of the best spells for surviving and Clarity restores mana once you run out. Having these will greatly help in both surviving and fighting since Flash is also good for an offensive use as well. It isn't too hard to survive long enough for Clarity too be finished cooling down after the first time you use it, if you are not soloing. Using Clarity after the first battle will allow you to keep fighting while the other enemies or even teammates either died or ran, therefore leading to a bit of free farming on 5v5 or it gives you the ability to wait in the bottom lane of 3v3 for the first wave of minions.

Alright now we have the pros and cons of this build.

Pros:
-Still gain a good amount of Ability Power while also not dying too fast.
-Rebirth's egg becomes harder to kill since it's defense is also based off of total -defense/magic res -25%
-Higher HP then most mages should really have
-Zhonya's Hourglass' Active can save you when Rebirth can't or is on cooldown.

Cons:
-This build doesn't work very good as a solo Anivia build in Summoner's Rift, almost forcing her to be in a lane with somebody.
-As this build gives less Ability Power then most builds for Anivia, therefore her burst damage is easier to counter.
-The build is rather expensive
-If it takes too long to get the Rod of Ages (20-22 minutes in 5v5, 18-22 minutes in 3v3 roughly speaking) you will not have much ability power mid game and will start falling behind, not coming back until near late game.
-If leveled exactly as I put it on here, Crystallize is not a great form of escape till late game.