What brand of bottled water is rated the highest in terms of purity? 
By purity I mean the safest level of minerals, no artificial
indgredients added, or the conditions in which the water is bottled
at.  I have seen some bottled water with sodium added and I have no
idea why.  I have heard Fiji water is ranked high, but with so many
brands advertising how they have the purest product, it's hard to
decide.

Are you familiar with Mountain Valley Water from Hotsprings Arkansas? 
My family in the Philadelphia PA area has been very pleased with it as
out sole source of drinking water for the past 25 years. Here's a link
to a helpful page on their website.
 http://www.mountainvalleyspring.com/faqs/
It is listed as sodium free, under NSF standards, has a PH level of
7.8 (moderately alkaline), and carries the approval mark of the NSF.
Personally I find the taste far superior to many well known French and
Italian waters. There is detailed info on the Mountain Valley website
of the breakdown of all the minerals that are naturally contained in
the water, and a description of the methods used to ensure purity
through the bottling process. Also, it's available in a wide variety
of glass bottles--and I tend to prefer the taste of water stored in
glass. Hope this is a little helpful.

('pure' water has a pH of 7 by definition)

Why anyone persists in paying for bottled water, when higher purity
(in the correct sense) comes out of the tap, beggars belief.

I think that Consumer Reports did some kind of study on bottled water
a couple years ago.  As I recall, their recommendation for most places
in the USA was to drink tap water.

The questioner asked about bottled water--not the merits of paying for
water or drinking it out of the tap. There are many individual reasons
why someone would need/prefer bottled water--ranging from poor taste
of the muncipal water to a compromised immune system. It seems
unnecessary to make a judgemental comment. Actually, even my house
plants die from my municipal water--and we need to use distilled
bottled water. Following your line of reasoning to the absurd, maybe I
should stick to house plants that can survive on only air--why waste
my tap water (which we pay dearly for here).

Ahhh come now.  The world would be horribly wishy washy if it wasn't
for judgements.

If I was immuno-compromised, I'd certainly not want to be drinking
unchlorinated water.

And tap purity is higher.

Darwin in action again I guess.

The Natural Resources Defense Council did a good study of bottled
water a few years back, you can access their report at
http://www.nrdc.org/water/drinking/bw/bwinx.asp

You really need to check the appendix for detailed information. 
http://www.nrdc.org/water/drinking/bw/appa.asp
Depending on what you are seeking, be it low mineral content, no
detectable heterotrophic plate counts, you may find a different result
for the "highest purity".

If you are really looking for the lowest mineral content, you would be
better off purchasing distilled water, or, more cheaply perhaps, a
distiller unit and making your own.  Note though that most low mineral
content waters tend to taste "funny" to many consumers, and also seem
to pick up a metallic taste.

Many bottled waters, like sodas, and beer, are bottled locally, and
thus depend a bit on the local water supply and to a lesser degree, on
the operations of the local bottler.  Aquafina and Dasani, for
example, do use local water supplies, but then add minerals back in
order to achieve a more consistent flavor, alkalinity, and mineral
content.

Other bottlers that focus on their source, such as Evian and Fiji, use
a single source and thus are probably fairly consistent whether you
buy then in NYC or or in Peoria.

Also note that the business of bottled water is changing rapidly, and
some of the players are getting bought up by some of the largest
international bottlers (Nestle Waters lists 10 brands under their
ownership including such notables as Perrier, Poland Springs and Ice
Mountain)

There are things in bottled water that were not there before they put
it in the bottle...solvents from the bottling system and residue of
heavy metal catalysts from the plastic...not to mention the
plasticizers in the plastic that play with our endocrine systems.  I
know you are looking for "tested and known results" but I don't think
the "all perfect" test exists.  I put tap water into glass juice
bottles but I should really filter it thru at least a charcoal filter
first.  But if your tap water tastes terrible then that is a cue not
to drink it. The quest for good water turns out to be a juggling act. 
Hulda Clark espouses tap water because even with its faults it does
not have the solvents from the bottling system and the hazards of
plastic.  I'd buy the Mountain Valley Spring Water as recommended
earlier if by budget would bear it.  There is no simple answer to what
we thought would be so simple...water!!