The invention relates to a high pressure method for producing pure melamine by pyrolyzing urea in a vertical synthesis reactor. The synthesis reactor has three stages above one another: a) in the first stage, the smaller portion of the total amount of urea is introduced into the central tube of a first tank reactor forming a first melamine-containing reaction medium; b) in the second stage, the first melamine-containing reaction medium and the larger portion of the total amount of urea is introduced into the central tube of a second tank reactor forming a second melamine-containing reaction medium; c) in the third stage, the second melamine-containing reaction medium is introduced into a vertical tubular flow reactor forming a raw melamine melt that is processed to obtain pure melamine.