With Veigar, taking mid is a good thing as you can feed you Ap withBaleful Strike faster.

Also, the site doesnt count Baleful passive and Mejai. This build get you between 900-1200 Ap if you feed a lot withBaleful Strike.

RushingTear of goddess at start of game insure your mana pool to grow a lot faster, finishing with 3500 Mana at lvl 18. As Veigar is mana-hungry early game, he needs a bigger mana pool.

TheSorcerer shoes then keep your speed the same as most character, giving you spell penetration.
Mejai's Soulstealer helps build a lot of AP as it gives you 2 stack each kill and 1 stack each assist *(Each stack being an 8 AP bonus and at 20 stack, you get cooldown reduction. You lose a third of your stacks when you die.)*

*As of here, I popElixir of brilliance as soon as I get back to the Summoner platform and I got 250 and can't get my next item.*

Comes nextMoonflair Spellblade which reduce crowd control effects on you by 25%. Pretty useful as Veigar needs to get back into fighting to deals a lot of damage. And YES it does reduce Silence time.

Then comesRabaddon Deathcap, with his 155 AP and his neat 30% more AP. *(Which is UNIQUE.)*

Then I finishArchangel Staff, as his passive is fed up from a while already so now I get some bonus AP from my mana increasing item. Also, it gives 3% of your max mana in AP which is massive with him.

I finish withLich Bane, starting with theAmplifying Tome to get more AP thenSheen, finishing Lich Bane for a price of 2210$ insuring a massive 900-1200 bonus damage to my auto-attack after I cast any of my spells.

I then pop Elixirs() until the end.

First, Veigar passive,Equilibrium, give 0.75% of mana regen for each 1% of mana you miss. Thus increasing the effectiveness of mana regen items.

TakingBaleful Strike at level 1, 3, 5, 7 and 9 is crucial as you quickly increase your damage with this spell, thus giving you easier time activating the passive of the spell: Each time you kill something (a minion or a champion) with the spell, you gain 1 AP permanently. Each champion kill give you (spell level) AP. *(This part work even if you kill with another spell.)*

TakingDark Matter at level 2 give you the possibility to harass your opponent by flinging a Dark Matter right in his minions, next to him. Shall it fear the hero to run or shall it hit him, the mana isn't wasted as you make the other player lose some time to feed. *(Dont waste any mana hitting minions only with this spell as Veigar is REALLY costly in mana in early game.)*

GettingEvent Horizon at level 4 give you an easy landing for Dark Matter or simply to stun your enemy to run away from a gank as ganks generally get called soon after mid usually hit level 5-6 (Under which you wont have the time to drop a point in Event Horizon.
Primordial Burst normally come at level 6, 11 and 16, no need to tell you why. *(Try not to use at low level until you get the chance to cripple or kill your opponent.)*

Then just alternate betweenDark Matter andEvent Horizon, dropping points in Baleful Strike at 9 and Primordial Burst at 11 and 16.

Upgrading Dark Matter increase your nuking damage output greatly.
Upgrading Event Horizon on his side, helps getting more time to nuke or run as the stun time and cooldown reduction upgrade with levels.