This route has more 4th and low 5th scrambling than climbing, but overcoming the "12 foot wall" will require a short 5.7 pitch.
Approach from either the east or west side as described on the Spire parent page. Note that the gully from the east, while still relatively clear of brush by Organ standards is getting more growth as the years pass. It also remains very loose in places.
From the saddle between the Spire and Low Horn #6, descend to the west slightly (there is a small bivy platform), climb up a series of small 4th/low 5th class shelves aiming to get yourself between The Spike and the Spire at a large cave like chimney crack. Either climb the crack, or climb some smooth slabs to the right to a perch where the NE face of The Spire comes into view (Ingraham says NW, but I believe that to be in error).
Work your way across a ledge in this face, over two awkward chimney moves behind blocks to a comfortable ledge below a steep 12 foot wall with a right facing corner crack. You may or may not decide to rope up for this section.
This little 12 foot wall is the crux of the route, and is also described in the North Face Direct route. The crack protects well, and leads you to the rappel anchors. From the rap anchors (or above), scramble (more 4th class) to the exciting summit.
Descent by retracing your ascent route, with one short rappel off of two old 1/4" bolts (rappel right, all the way past the chimney moves you made crossing the ledge on the way up). Most of the down climbing that you came up is easier than it looks from above, especially getting down to the rap anchors.
Light rack, of medium cams or large wires.
Two old 1/4" bolts for rappelling the crux.
You may find rappel slings in various locations as different parties may choose to rappel some of the 4th class sections to avoid down climbing them.
We used a 'doubled up' 70m twin rope (i.e. 35m), and had plenty to climb the crux pitch and to rappel safely down the crux and past the two chimney sections on the ledge.