diff --git a/internal/auth/session.go b/internal/auth/session.go
index 3a93a70..89d061e 100644
--- a/internal/auth/session.go
+++ b/internal/auth/session.go
@@ -93,10 +93,21 @@ func (m *SessionManager) StartSession(w http.ResponseWriter, r *http.Request, cl
 	// The cookie outlives idle gaps so a returning device presents its ID;
 	// the server decides whether the session is still live. Max-Age is whole
 	// seconds, rounded up so the cookie never ends before the absolute limit.
-	http.SetCookie(w, sessionCookie(id, int((m.max+time.Second-1)/time.Second)))
+	http.SetCookie(w, sessionCookie(id, cookieMaxAge(m.max)))
 	return nil
 }
 
+// cookieMaxAge is lifetime in whole seconds, rounded up. It rounds the
+// quotient rather than adding to lifetime, which would overflow near the
+// largest time.Duration.
+func cookieMaxAge(lifetime time.Duration) int {
+	seconds := lifetime / time.Second
+	if lifetime%time.Second != 0 {
+		seconds++
+	}
+	return int(seconds)
+}
+
 // SessionFor returns the live session r carries and slides its last use to
 // now. It returns ErrNoSession when there is none; an expired session is
 // deleted.
diff --git a/internal/auth/session_test.go b/internal/auth/session_test.go
index 859cf3b..20e03fe 100644
--- a/internal/auth/session_test.go
+++ b/internal/auth/session_test.go
@@ -333,6 +333,22 @@ func TestSessionCookieOutlivesFractionalMax(t *testing.T) {
 	}
 }
 
+// A lifetime near the largest accepted duration still yields a positive,
+// rounded-up cookie Max-Age rather than overflowing.
+func TestSessionCookieMaxAgeNearLargestMax(t *testing.T) {
+	t.Parallel()
+	largest, err := time.ParseDuration("2562047h47m16.854s")
+	if err != nil {
+		t.Fatal(err)
+	}
+	h := newSessionHarness(t)
+	h.sessions.max = largest
+	_, cookie := h.signIn(t)
+	if want := 9223372037; cookie.MaxAge != want {
+		t.Errorf("session cookie MaxAge = %d, want %d", cookie.MaxAge, want)
+	}
+}
+
 // A lifetime shortened by configuration applies to existing sessions even
 // though their stored expires_at is later.
 func TestSessionLifetimeChecksUseCurrentConfig(t *testing.T) {