Is anyone willing to have a go at a rewrite?
Here's my attempt, Tsyganka. I decided truth might be one of the few things nearly as unwelcome as death, sometimes.
I struggled with cornice, but decided to leave it, for lack of a better option. Gutter could at least be near the same location, but it is such an ugly word! Garden fitted the intent better, but is too far from the meaning of cornice.

Because I could not stop for Truth,
He kindly stopped for me;
The car held only just ourselves,
And Immortality.

We slowly drove, there was no haste,
And I had put away
My brash career, and leisure too,
For his serenity.

We passed the school where children played,
Their lessons scarcely done;
We passed offices newly plain,
We passed the setting sun.

We paused before a house that seemed
A swelling of the ground;
The roof was scarcely visible.
The cornice but a mound.

Since then 'tis centuries; but each
Feels shorter than the day
I first surmised the bright headlights
Were toward eternity.