Is it possible to complete Dark Souls without leveling?

Yes it is, there are several things in the game that make you stronger without affecting level.

1. Pyromancy is a spell class that levels completely based off the items and the souls you spend on enhancing it. In fact the magic adjust of pyromancy before the nerf was the highest among all spell trees even with a 50 INT Sorcerer. Currently it ranks as high as around 40 INT at max level, or possibly higher. The damage is completely independent of level and as such I have personally made a Lv15 Character with max pyromancy who has slain the four kings.

2. Equipment can dramatically assist in the damage a character can do. A low level character wouldn't have the endurance to wear a lot of heavy armor but further down I explain how that can be mitigated to. Even with low endurance almost any armor that is fully enhanced is extremely powerful. For weapons, the elemental weapons like Fire and Lightning do not scale at all but have fantastic base damage making them very powerful and equip-able on Lv10 characters. Even a basic longsword put to +10 Fire is extremely deadly.

   In addition much equipment boosts stats, mask of child or father both provide bonuses like hp increase or weight load increase, the grass crest shield increases stamina regeneration.

3. Rings can mitigate a lot of the need to level. The ring of favor off of lautrec increases HP, Weight Load and Stamina by a considerable amount at the cost of never being able to take it off without destroying it. The havel ring dramatically increases weight load, the physical ring can make up for wearing a lot of light weight armor by increasing physical defense by 50-70.

The combination of all three allow you to take very low level character and make them more than strong enough to down any boss in any scenario.

Also if you mean to do it without using any souls at all, that is certainly possible as well since nearly all attacks can be dodged but it would take a considerable amount of time to do it with no souls and require absolutely no mistakes however this game is still a RPG and a no souls run would be comparable to the nsgnsncnonennenbb runs of FFX.

Finally the game still takes plenty of opportunities to kill you regardless of stats by traps and falls.

I'm mid-way through a Soul Level 1 playthrough, using the Pyromancer class of course (since it's the only class that starts at SL1).

The only rules I've chosen for the playthrough is that I'm not allowed to increase my Soul Level, and I'm not allowed to summon human players. Upgrading weapons, armour, shield, pyromancy flame are all okay, plus I've also learned a few sorceries and miracles. I am limited in my choice of items however, since I can't upgrade my stats to meet the requirements of many of them (eg. I can't use the Drake Sword or any type of spear), and I only have 2 slots for magic/pyromancy spells. Summoning NPCs is okay.

The last couple of times I played using this character I defeated the Bell Gargoyles, the Capra Demon and the Gaping Dragon. I'm currently somewhere in the middle of Blighttown.

I wouldn't say I'm having any more difficulty with the game than I did on my first regular playthrough. I should be able to finish the game eventually.

UPDATE: I've now completed Anor Londo, obtaining the Lordvessel, and have also defeated the Four Kings boss. Ornstein & Smough presented quite a challenge, but I figured them out in the end!

Yes. I'm on a soul lvl 10 that I have completed the game with 5+ full catarina and lightnig weps maxed. It was a lot of fun stopping at 10 and playing through.