Which cut of beef has the least amount of fat? (loin, rib , round, etc.)
While it is not generally considered a cut, the heart has no fat tissue at all between the muscle fibres (the marbling in skeletal muscle). Depending on how it is trimmed, it can have a significant amount around the outside of the atrioventricular junction, but this can be removed prior to cooking fairly easily.
At my butchers it is the cheapest muscle available. When roasted it is delicious, as the structure holds in much more of the juices than more traditional cuts. It is worth bearing in mind that it does not fall apart with long cooking times as the structure is not maintained by collagen but the branching structure of the muscle cells, so has a very different texture in stews.
Beef round is the leanest generally available cut. It's the cut which is most commonly used for dried beef (where you want a low fat content).
Beef shank is also low in fat if carefully trimmed. It is an extremely tough cut, and may be difficult to find in Western butcher shops.
There are various other, smaller cuts which are also quite low in fat, such as filet mignon and skirt steak. These are likely to be expensive and/or difficult to find.