Using exact enumeration methods and Monte Carlo simulations, we study the phase diagram relative to the conformational transitions of a diblock copolymer in two dimensions. The polymer is made of two homogeneous strands of monomers of different species which are joined to each other at one end. We find that, depending on the values of the energy parameters in the model, there is either a first order collapse from a swollen phase to a compact phase of spiral type, or a continuous transition to an intermediate zipped phase followed by a first order collapse at lower temperatures. Critical exponents of the zipping transition are computed, and their exact values are conjectured on the basis of a mapping onto percolation geometry, thanks to recent results on path-crossing probabilities.