diff --git a/crates/epaint/src/text/text_layout.rs b/crates/epaint/src/text/text_layout.rs
index 276f3f9e2a3..7f1cd865710 100644
--- a/crates/epaint/src/text/text_layout.rs
+++ b/crates/epaint/src/text/text_layout.rs
@@ -238,6 +238,12 @@ fn rows_from_paragraphs(
 }
 
 fn line_break(paragraph: &Paragraph, job: &LayoutJob, out_rows: &mut Vec<Row>, elided: &mut bool) {
+    let wrap_width_margin = if job.round_output_size_to_nearest_ui_point {
+        0.5
+    } else {
+        0.0
+    };
+
     // Keeps track of good places to insert row break if we exceed `wrap_width`.
     let mut row_break_candidates = RowBreakCandidates::default();
 
@@ -253,7 +259,7 @@ fn line_break(paragraph: &Paragraph, job: &LayoutJob, out_rows: &mut Vec<Row>, e
 
         let potential_row_width = paragraph.glyphs[i].max_x() - row_start_x;
 
-        if job.wrap.max_width < potential_row_width {
+        if job.wrap.max_width + wrap_width_margin < potential_row_width {
             // Row break:
 
             if first_row_indentation > 0.0
@@ -630,7 +636,24 @@ fn galley_from_rows(
         num_indices += row.visuals.mesh.indices.len();
     }
 
-    let rect = Rect::from_min_max(pos2(min_x, 0.0), pos2(max_x, cursor_y));
+    let mut rect = Rect::from_min_max(pos2(min_x, 0.0), pos2(max_x, cursor_y));
+
+    if job.round_output_size_to_nearest_ui_point {
+        let did_exceed_wrap_width_by_a_lot = rect.width() > job.wrap.max_width + 1.0;
+
+        // We round the size to whole ui points here (not pixels!) so that the egui layout code
+        // can have the advantage of working in integer units, avoiding rounding errors.
+        rect.min = rect.min.round();
+        rect.max = rect.max.round();
+
+        if did_exceed_wrap_width_by_a_lot {
+            // If the user picked a too aggressive wrap width (e.g. more narrow than any individual glyph),
+            // we should let the user know.
+        } else {
+            // Make sure we don't over the max wrap width the user picked:
+            rect.max.x = rect.max.x.at_most(rect.min.x + job.wrap.max_width);
+        }
+    }
 
     Galley {
         job,
diff --git a/crates/epaint/src/text/text_layout_types.rs b/crates/epaint/src/text/text_layout_types.rs
index c14348ecf29..fde36ec6a50 100644
--- a/crates/epaint/src/text/text_layout_types.rs
+++ b/crates/epaint/src/text/text_layout_types.rs
@@ -74,6 +74,10 @@ pub struct LayoutJob {
 
     /// Justify text so that word-wrapped rows fill the whole [`TextWrapping::max_width`].
     pub justify: bool,
+
+    /// Rounding to the closest ui point (not pixel!) allows the rest of the
+    /// layout code to run on perfect integers, avoiding rounding errors.
+    pub round_output_size_to_nearest_ui_point: bool,
 }
 
 impl Default for LayoutJob {
@@ -87,6 +91,7 @@ impl Default for LayoutJob {
             break_on_newline: true,
             halign: Align::LEFT,
             justify: false,
+            round_output_size_to_nearest_ui_point: true,
         }
     }
 }
@@ -180,6 +185,7 @@ impl std::hash::Hash for LayoutJob {
             break_on_newline,
             halign,
             justify,
+            round_output_size_to_nearest_ui_point,
         } = self;
 
         text.hash(state);
@@ -189,6 +195,7 @@ impl std::hash::Hash for LayoutJob {
         break_on_newline.hash(state);
         halign.hash(state);
         justify.hash(state);
+        round_output_size_to_nearest_ui_point.hash(state);
     }
 }