It is conjectured  (see e.g. Goldston-Pintz-Yildirim http://arxiv.org/abs/1103.5886) that 

$$
 \lim_{n \to \infty}\frac{1}{n}\#\Big\{m \le n : \alpha<\frac{p_{m+1} - p_m}{\ln p_m} < \beta \Big\} = \int_\alpha^\beta e^{-t} dt,
$$
and thus in particular your value should be $1-e^{-1}$. (GPY proved that this is positive for any $0=\alpha<\beta$). I am not sure about recent progress (in particular if Zhang's recent breakthrough result is relevant) on this problem.