It is shown that for each 1 < p < 2 there exist identically distributed uncorrelated random variables $X_n; with;E({$mid$X_1$mid$}^p);<;{infty}$</TEX>, not fulfilling the weak law of large numbers (WLLN). If, however, the random variables are moreover non-negative, the weaker integrability condition $E(X_1;log;X_1);<;{infty}$ already guarantees the strong law of large numbers.