The central limit theorem (CLT) and law of large numbers (LLN) are both statements about the sample mean. If the sample mean of $n$ samples is $\bar{X}_n$, the CLT says that the distribution of:
$$\frac{\bar{X}_n-\mu}{\sigma_{\bar{X}_n}}$$
where $\mu$ is the true mean of the samples, $\sigma$ is their true standard deviation and $\sigma_{\bar{X}_n} = \frac{\sigma}{\sqrt{n}}$, is a standard normal random variable when $n$ is large.
The law of large numbers says that $\bar{X}_n$ becomes closer to $\mu$ as $n$ becomes large.
Isn't then the law of large numbers just a weaker version of the central limit theorem?
No, it is not, but for most cases you are right. If the variance is infine, CLT doesn't really mean anything, but LLN still works.
For an example, suppose your sample space is the square roots of every power of two (with positive integer powers), and their distribution is defined by $P(X = \sqrt{2^k}) = 2^{-k}$. You can easily show that this determines a distribution with finite expectation (the expectation is bounded, as $\sum_{k = 1}^\infty 2^{-\frac{k}{2}} \leq 2\sum_{k = 0}^\infty 2^{-k} = 4$), but an infinite variance, as $E(X^2) = \sum_{k = 1}^\infty 1 = \infty$.
For this example, LLN works, CLT does not.