To cope with the 40 MHz event production rate of LHC, the trigger of the ATLAS experiment selects the events in three sequential steps of increasing complexity and accuracy whose final results are close to the offline reconstruction. The Level-1, implemented with custom hardware, identifies physics objects within Regions of Interests and operates a first reduction of the event rate to 75 KHz. The higher trigger levels provide a software based event selection which further reduces the event rate to about 100 Hz. This paper presents the algorithm (muFast) employed at Level-2 to confirm the muon candidates flagged by the Level-1. muFast identifies hits of muon tracks inside the Muon Spectrometer and provides a precise measurement of the muon momentum at the production vertex. The algorithm must process the Level-1 muon output rate (~20 KHz), thus a particular care has been used for its optimization. The result is a very fast track reconstruction algorithm with good physics performances which, in some cases, approach those of the offline reconstruction: it computes the pT of prompt muons with a resolution of 5.5% at 6 GeV and 4.0% at 20 GeV and with an efficiency of about 95%. The algorithm requires an overall execution time of ~1 ms on a 100 SpecInts95 machine