As @IrishStat said it depends on the model. One way of recovering constant is to use the mean value of the residuals. Note that this method relies strongly on certain assumptions. Here is the illustration. Suppose your model is
$$Y_t=\alpha + \beta X_t + \varepsilon_t$$
and you estimate it by
$$\Delta Y_t=\beta \Delta X_t+\Delta \varepsilon_t.$$
Suppose your estimate $\hat\beta$ is unbiased (or at least consistent). Define
Substituting the true model we have
if $\hat\beta$ is unbiased or
if $\hat\beta$ is consistent.
So the natural estimate for $\alpha$ is
Note that the model assumption is critical here. However with care this trick can be applied in general.