Recent mode coupling theory (MCT) calculations for a hard‐sphere system with a short‐range attraction show that one may observe a new type of structurally arrested state originating from clustering effect, called the “attractive glass”, as a result of the attractive interaction. This is in addition to the well‐known glass‐forming mechanism due to the cage effect in the hard sphere system, called the repulsive glass. The calculations also indicate that, if the range of attraction is sufficiently short compared to the diameter of the hard sphere, within a certain interval of the volume fraction and the effective temperature, the two glass‐forming mechanisms can compete with each other. For example, by varying, the effective temperature at appropriate volume fractions, one may observe respectively, the glass‐to‐liquid‐to‐glass re‐entrance or the glass‐to‐glass transitions. Here we present experimental evidence for both transitions, obtained from small‐angle neutron scattering (SANS) and photon correlation spectroscopy (PCS) measurements taken from denseL64 copolymer micellar solutions in heavy water. We show, by varying the temperature in the predicted volume fraction range triggers a sharp transition between the two types of glass. In particular, according to MCT, there is an end point (calledA3singularity) of this glass‐to‐glass transition line, beyond which the long‐time dynamics of the two glasses become identical. Our findings confirm this theoretical prediction. Surprisingly, although the Debye‐Waller factors (DWF), the long‐time limit of the coherent intermediate scattering functions, of these two glasses obtained from PCS measurements indeed become identical at the predicted volume fraction, they exhibit distinctly different intermediate time relaxation. Furthermore, our SANS results on the local structure obtained from volume fractions beyond the end point are characterized by the the same features as the repulsive glass obtained before the end point. A complete phase diagram giving the boundaries of the structural arrest transitions forL64 micellar system is given. © 2004 American Institute of Physics