The block fading (BF) channel is a useful model for various communication systems in urban environments. Full-diversity error-correcting codes are required to approach the physical limits of the performance on BF channels. Low-density parity-check (LDPC) codes are good error-correcting codes, but full-diversity standard random LDPC codes for the BF channel are not known yet. We design full-diversity random LDPC ensembles at infinite block length by optimizing the threshold in multiple points in the fading space, which takes into account the randomness of the fading. However, this is not sufficient to achieve full-diversity at finite block length, because of stopping sets. We therefore propose a method to generate full-diversity code instances at finite length so that stopping sets over information bits are avoided. The asymptotic and finite length word error rate performance is verified by means of density evolution and Monte Carlo simulations, respectively, confirming that full-diversity standard random LDPC codes exist and perform well.