A new Monte Carlo algorithm for the simulation of atomistically detailed polymer melts is presented. The method introduces connectivity relationships as variables in the description of the polymer. The connectivity of the polymer is altered in Monte Carlo moves that satisfy the detailed constraints of molecular geometry. Connectivity-altering moves are seen to induce large jumps in the configuration space of the bulk polymer, thereby greatly enhancing the efficiency with which molecular configurations are sampled. Simulations are carried out in a semigrand ensemble in which the chain length distribution is controlled by a spectrum of chemical potentials. Limiting chain length distributions are derived and compared with simulation results. Volumetric and structural predictions of the method are found to be in agreement with previous work.