jmc

Let's denote the first term as $a$ and the common ratio as $r.$ Additionally, call the original sum of the series $S.$ It follows that $$\frac{a}{1-r}=S.$$ After the first three terms of the sequence are removed, the new leading term is $a{r}^3.$ Then one $27^{\text{th}}$ of the original series is equivalent to $$\frac{a{r}^3}{1-r}=r^3\left(\frac{a}{1-r}\right)=\frac{S}{27}.$$

Dividing the second equation by the first, $r^3=\frac{1}{27}$ and $r=\boxed{\frac{1}{3}}.$